If iron can’t undergo fusion, does that mean a black hole is mostly iron? Since stellar fusion can’t progress beyond iron, and a large enough star collapsed into a black hole because an iron core stalled fusion, wouldn’t that mean all black holes are predominantly iron?
 A: If we are talking about stellar-sized black holes, then the object that collapses to form a black hole will have a high concentration of iron (and other iron-peak elements like manganese, nickel and cobalt) at its core, and it is the core-collapse that begins the black hole formation process, but much more material than this will eventually form that black hole.
It appears, empirically, that the minimum mass of a stellar-sized black hole is around $4M_{\odot}$, but is more typically around $10-15M_{\odot}$. But the extinct core of iron in a pre-supernova star is unlikely to exceed around $1.5-2M_{\odot}$ even for the most massive of supernova progenitors (see for example these slides).
Thus most of the material that collapses into a black hole is not iron, it is actually the carbon, oxygen, silicon neon and helium that surrounded the iron core. Much of the nuclear material will be photodisintegrated into its constituent baryons (or alpha particles) during the collapse. Neutronisation reactions will turn most of the protons in the high density material into neutrons. Even at equilibrium, when densities higher than about $10^{14}$ kg/m$^{3}$ are reached then any remaining nuclear material will begin to transmute into all sorts of weird and wonderful neutron-rich nuclei (as in the crusts of neutron stars) and by the time you reach densities of $\sim 10^{17}$ kg/m$^{3}$ (which is still well outside the event horizon of a stellar-sized black hole), the nuclei will lose their identity in any case, and become a fluid of neutrons, protons and electrons.
A second point to consider is whether it makes any sense to talk about the composition of a black hole. Composition is not one of the things you can measure - these are restricted to mass, angular momentum and charge. The other details are lost from a (classical) black hole.
A: Iron can undergo fusion. However, iron is the point where fusions starts to cost more energy than it yields, so in a typical star it doesn't fuse.
In a supernova, and the abundance of energy available in one, iron will continue to fuse to heavier materials, which is probably how we got heavier metals here on earth in the first place (it has to have fused somewhere, after all).
More to the point, in the case of black holes, when they collapse at first, see Rob Jeffries' answer, because he explained that bit better than I could.
After it is a black hole, it becomes impossible to talk about its composition in terms of mass and energy that we are used to. While we're not sure about electrons (AFAIK), protons and neutrons have a hard diameter. Removing the space between them causes for example a neutron star, which you could consider one very, very large atom, from a certain point of view:

Most of the basic models for these objects imply that neutron stars are composed almost entirely of neutrons (subatomic particles with no net electrical charge and with slightly larger mass than protons); the electrons and protons present in normal matter combine to produce neutrons at the conditions in a neutron star. Neutron stars are supported against further collapse by neutron degeneracy pressure, a phenomenon described by the Pauli exclusion principle, just as white dwarfs are supported against collapse by electron degeneracy pressure.
–"Neutron star", Wikipedia

However, at a certain pressure, even the neutrons/protons cannot uphold more matter that way. That's when we get black holes. We know they're more compact than neutron stars, but we don't even know how big they are - only that they're smaller/equal to their event horizon. We don't even know if they HAVE a size.
So black holes are really two steps behind of where you could call a bunch of matter "iron". Step 1 is the neutron star, where we already lost all information about what those particles USED to be before it was one. And step 2 is the black hole, where we don't even know what those particles/matter/energy currently are. Probably not anything we've seen so far.
Most of the essentials here can be found at Wikipedia:


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*Black hole

*Degenerate matter
A: There is a missing step or two which is causing your confusion.
I'm going to focus on what happens in a huge star with an iron core, and during a collapse into a black hole, which will help to explain the answer much better.
The core of a big star
In an "ordinary" star, the particles in its core are a mixture called a plasma - their electrons stripped and free, and the nuclei free as well. Stars are pretty big, and in the core of a star, with that much pressure and temperature, the nuclei of atoms can occasionally combine to form larger nuclei. Because the larger nuclei are more stable (up to a point), when nuclei fuse like this, the larger nucleus they create needs less energy to hold it together - called "binding energy" - than the binding energy of the two or more nuclei that created it. This extra energy which it "doesn't need" any more, is given off, and that's how the star generates energy and powers any light or heat ("radiation") that it gives off.
Now, like a lot of things, this fusion process has a limit. A bit like a discount at the superstore - buy one item, it's £1. Two items might be £1.70 (=£0.85 each), 3 items £2.40 (=£0.80 each)... but eventually the discount thins down and buying a thousand doesn't really get you more of a discount per item. It's a crude analogy but fusion works the same. You get a lot of energy by combining two hydrogen nuclei to make a helium nucleus. Quite a lot from combining helium. By the time you're combining silicon to get nickel and iron in a large star, you just can't get more energy out. It's as stable as a nucleus can ever get, and more combining can't be more stable. So iron literally has nowhere to go. Once the star's core is predominantly iron/nickel, the processes of fusing which have powered it, for all its life, simply stop, in any practical sense. They can't  fuse to more stable elements, so they just don't fuse.
(Technical comment - fusion doesn't exactly stop, it just becomes statistically much less likely to drive any specific direction or produce any net energy. Any fusion that gives rise to elements above iron is tiny compared to the chances of those same elements above iron quickly breaking down again. The reactions now absorb energy to create heavier elements, which have a higher chance of breaking down and releasing that energy again before heavier elements can build up. So the net effect is that fusion stops, in the sense that in in an iron core these processes go one way as much as the other, and ends up going nowhere overall.)
Now let's look at the star as a whole. All its life, the star has been trying to collapse inward due to the force of gravity. But it really hasn't. Why not? Two reasons. 


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*Heat / expansion - First, the core is immensely hot, and that heat tends to resist the star collapsing. As the star collapses, the centre compresses, so it heats up, fusion goes faster, so it gets hotter, so it tries to expand again. So a balance is struck, it can't collapse. 

*Exclusion principle / degeneracy pressure - The second reason it doesn't collapse is a strange quantum mechanics principle called the "Exclusion Principle" - basically if you try to squeeze two subatomic particles called "fermions" together, they resist. Nuclei act like this and they also resist being squeezed tightly together, and so do electrons. This is actually why matter takes up space around us - the particles are held apart by their own quantum nature. Same goes for the core of a big star. The force that the exclusion principle appears to exert is called "degeneracy pressure".
The core collapses and everything goes crazy
But now, the big star's core has fused all it can into iron, so what happens next? Gravity still tries to make it collapse, but now, even when it is collapsing and compressed, it can't result in more fusion and more heat - it's fused all it can do, and can't fuse any more. So suddenly, all that is left is the degeneracy pressure from the exclusion principle, standing alone, to resist the pressure inwards - and it can't. It has a limit and in a star of the kind we're talking about, its best just isn't enough. Suddenly, like a wall giving way, the entire resistance against collapse that the exclusion principle was giving to the core, gives way and fails. 
Now there's nothing at all to fight gravity, and with no resistance, the inner part of the star collapses inward at a startling speed. It reaches a speed of 3/4 of the speed of light in just a tiny fraction of a second as the star's entire core collapses inward towards its centre with no resistance at all.  The collapse does eventually halt in some cases - see below - but only when the core has almost reached a tiny size just a few miles across.
Inside the collapsing core, particles are interacting at an insanely energetic level. New nuclei are born that couldn't be formed in any less a furnace than the final milliseconds of life of a dying star (this is a major source of many of the elements heavier than iron). 
After the core collapses
Within the collapsing core, however, something is happening that might halt the collapse. Protons and electrons are rapidly forced to combine into neutrons - and the force holding neutrons apart is far greater than the degeneracy force in the original core. It's sometimes called neutron degeneracy pressure but more correctly it's almost always the strong force not the exclusion principle that's kicking in when the neutrons get really close together, and which holds them apart.  Either way, it's that pressure between neutrons which finally halts the collapse in all but the largest giant stars.
At that point, one of 3 things can happen:


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*Ordinary big star - If the star is up to a certain size, it will collapse to the point that part of the inner core is converted to neutrons with virtually no space between them (a "neutron star"), at which point it's as if our original wall, that had suddenly vanished, suddenly reappears - and comes back as several miles thick mass of concrete. The outer core is falling in at 3/4 speed of light, and (ignoring some handwaving about stalled collapse and neutrinos) ..... it suddenly hits the universe's most solid concrete block. Bang. Rebound. Biggest. Explosion. In. The. Universe. (pretty much). The technical name is a Type 2 (or "core collapse") supernova - there are other ways that a supernova can happen in different kinds of stars, but I'm not going there in this answer.  For a short while, the core gives out enough energy that (as XKCD famously (almost) said), if you ignore the protective effect of earth's magnetic field and atmosphere, you'd get a billion times more of a radiation dose from a supernova as far away as the sun, than from an H-bomb resting on your eyeball. Or as one website puts it, the energy given out by a core collapse  supernova within a few tens of light-years would destroy a large part of life on earth (there arent any big enough stars close to us, so don't worry!) That kind of "big". Insanely big. The outer layers of the star, which have been hanging around pretty much as usual during all of this, are suddenly shredded and ripped from the star by the sheer force of the rebounding outer core and flung into space, as a rapidly expanding cloud called a "nebula".

*Very big big star - If the star is very * very * big, even a neutron core can't stop collapse. The core never forms a neutron star, never rebounds, never explodes outward, because even the pressure between neutrons isn't enough to resist that much gravity. The star collapses to a point that we don't even know for sure what happens - there seems to be nothing that can stop collapse down to a literal single point in space. A black hole is born.

*In between the two - it tries to do both. A neutron core forms, there's a rebound...but it's weak. Gravity is too strong, and instead of dispersing in space, the rebounding matter begins to slow... and falls back onto the neutron core. The core gets bigger from returning material... bigger... and finally when enough mass has returned to the core, the balance tips, gravity wins, neutron degeneracy pressure gives way, and the inward collapse kicks off again. This time nothing can stop it, so... black hole again.


Now, this isn't exact. It misses out the possibility of quark or other degeneracy pressure, because that hasn't been proven yet. It ignores huge stars and pair production or other ways they can end their lives. It ignores some smaller stars that can collapse due to electron removal or explode without iron core collapse as type 1a.
But it gives an idea of the usual way a big star dies, why a big star that can't fuse iron, can still collapse to a black hole, and what goes on when this happens.
So ... is a black hole made of iron?
So finally, we can come back to your question: Is a black hole made of iron?
I think you can see the answer. 
The star's core was iron, but when the core collapsed, it stopped being iron. Some of the matter in the core was ripped apart, and converted to a flood of other particles and energy. Some of it naturally was still iron when it was flung into space. The rest of the core collapsed at a subatomic level, and isn't an element (like silicon or hydrogen or iron) at all, any more. It's a solid/liquid plasma of degenerate neutrons and neutron matter (and probably a ton of other states and particles as well), seething at a temperature of up to a thousand billion degrees in the immediate aftermath of being born as a neutron star. If that sounds exotic and complicated - it is!!  
But for the largest collapsing stars, when the neutron core collapsed to a black hole, even this structure was lost, it just became undefinable - an infinitely dense amount of matter in an infinitely small point of space. 
(Technical comment - physicists suspect that quantum processes mean it's "almost infinite" density in "almost infinitesimal" space, but for our purposes there isn't any real difference)
That's not iron. In fact, it's no element at all. It's not even neutrons or seething degenerate matter. It's just a black hole - an object defined by how massive it is, and how fast it's rotating (if at all: technically its angular momentum) and its electrical charge (if any).  It's way, way, not iron any more, and never will be.
Quick black hole note at the end
Although I've only described a black hole created by core collapse, the same will be true for black holes that are formed in any other way. No matter how it forms, the last 2 paragraphs also apply to every kind of black hole in the universe. 
What this means is that all black holes - however they were caused and whatever they started out as - have no internal structure we can ever know of, they retain none of their original elements, no neutron matter. 
(Technical comment: this isn't quite true, as in matter crossing the event horizon of a supermassive black hole, but it might as well be true as far as anyone outside the event horizon - defined below -  is concerned, and it's true for all practical and physical purposes, because nobody outside the event horizon can ever detect any difference anyway.)
All that black holes have, in terms of physical properties, is what a core collapse black hole also has - their mass, their rotation/angular momentum, and their charge, and in most cases (or perhaps all? we don't know for sure which) a limit called the "event horizon" at some distance from them all around, that dictates how close to them an object can get before it becomes physically impossible to see it, detect it, or interact in any way with it - because it's too close to the centre of the black hole for any light or other signals to escape the immense pull of gravity. 
Once any matter or energy crosses this event horizon, for all practical purposes it can be treated as losing whatever identity or contents it had, and just adding to the black hole. Inside the black hole, the closer it gets to the singularity at the centre, the more accurate this is (although we can never see it). At the centre itself, it is   correct as far as we know right now... but that's cutting edge stuff and nobody's absolutely clear what exactly happens at the singularity itself.
