Is a suit that hides a soldier's heat signature fundamentally possible? I recently played "Crysis", a game where the protagonist wears a suit that allows the player to hide both himself and his heat signature. Then I watched Iron Man 3, where a kid suggests that Tony Stark should have implemented retro reflection panels in his suit.
So I'm thinking, well, as is the nature of things, people are going to be pursuing this sort of thing in real life too, pretty soon. But I'm trying to figure out whether a suit can contain a person's heat signature without emitting the heat somewhere. Is such a thing fundamentally possible to do, without over-heating the person within?
 A: You could try to develop a material that acts like a fluorescent one, that is, transform infrared radiation into lower energy photons, such as microwave. So you will not glow on infrared but on some other wavelength of your choice. Now, if this is technologically feasible (using nano-engineered materials perhaps), I have no idea!
A: There are some nice ideas in other answers, but they are overseeing some basics. Let's do some thermodynamics. The efficiency of a thermal engine is bounded by the Carnot efficiency:
$$\eta \le 1 - \frac{T_c}{T_h} $$
Where $T_c$ is the temperature of the cold end and $T_h$ the heat source. Assuming we are in a cool environment, $T_c=0 C$, $T_h=37 C$, so: 
$$\eta_C \simeq 12\%$$
Whatever you do to convert your body heat into something useful, you will still have to pump out 88% of what you produce. And this is a very optimistic estimation, given that you have to make it portable. Way to go.
Let's look at the heat sink option. A human body has a surface of roughly $2 \mathrm{m}^2$, so it radiates about $10^3 \mathrm{W}$, but we are getting in around $600 \mathrm{W}$ from our surroundings (again, assuming a temperature of 0 C). If we leave base with a heatsink of iron cooled with liquid nitrogen (76 K) we have:
$$\frac{c_e \Delta T}{P}=\frac{449\frac{J}{K kg} 234 K}{400 W} = 262 \mathrm{\frac{s}{kg}} \simeq 4 \mathrm{\frac{min}{kg}}$$
So if we manage to redirect all our excess heat to the sink we will finish it in a few minutes.* A full hour of invisibility would require loading an extra 15 kg of weight. Other metals have a better specific heat ratio, but expect no wonders here.
How to actually become IR invisible
In order to become truly invisible, what we need is to dissipate all our heat via conduction. The way I would do it (but I am not an expert), is a multi layered system, where the inner layers are designed to give away heat, while the outer layers have to allow for ventilation while shielding the IR emited by the dissipation layer. The peak emission for a human body is in the range of $ 9 \mu m$, so a metallic mesh of that width should be a pretty good shield.
Now, you "just" have to figure out how to build a cloth with that metal embedded, while being able to endure hard conditions. But that is mere Engineering, not Physics.
In this scenario, all our heat is being carried away by the air around us, but that will not create a strong signal, as it is not dense. Also, turbulence will quickly mix it with the surrounding air, cooling it instantly.

*- The specific heat varies with temperature, but as a ballpark it works.
A: No, it is not possible to hide a person's heat signature indefinitely. Even with the best suit imaginable, you will eventually either begin leaking the heat, overheating the person, or both.
One problem is that there are no perfect thermal insulators. This means that you must either use the best available and keep emissions below some threshold of detection, or you must use active cooling of the surface of the suit (think heat pumps dumping heat inward) to hide. Both will eventually increase the internal temperature until something (quite likely the wearer of the suit) undergoes some form of thermal failure.
With that said, you can do tricks to hide thermal signatures better.
One trick is to create some kind of internal heat sink -- think of is as sort of a "cold battery," e.g. a block of ice -- that can extend your total time before heat failure happens. It can be bulky, but if done right, you can get very good hiding for a while.
The second trick is to convert the heat and radiate in some fashion that, at least for the target use, is hard to detect. One example (not necessarily a very practical one) would be to attach a highly efficient laser to the top of the suite and send heat off in the form of a very closely controlled laser beam, presumably at some frequency unlikely to be detected, and with careful aim to avoid hitting detectors.
There are all sorts of variants of that last idea. One amusing one would be "heat eggs" that you could drop as you go. At first they would look cool, but then the interior high heat that they have stashed away would start to show. The trouble with that idea is that you'd end up setting a trail of bread crumbs to the user!
If you want to get really exotic, you could in principle try to figure out a way to radiate heat in some extremely hard-to-detect form such as neutrinos. If you could use particles that hard to detect to radiate heat, the result would be pretty close to the ultimate "true stealth" heat suit.
2014-10-25.0841 EST - Addendum
First, @ThomasPornin pointed out in the comments that Kim Stanley Robinson used the heat egg idea in his Mars series, which I've read and clearly (if unconsciously) must have picked up the idea. A truly amazing bit of hard science fiction, that series!
Second, there are a few more ways to dump energy than vibrations (heat and sound), electromagnetic (light in all forms), or matter dumps (which would include both neutrinos and heat eggs). You could in principle at least also use fields, including magnetic, electric, and potentially even gravity. (The strong and weak forces are too short range for dumping.)
Magnetic induction comes to mind as the most plausible, since you can come up with real scenarios of energy transfer with that one, even if they would be hard to use in most field situations.
Electric fields are... an interesting possibility, but pretty hard to quantify? I'm thinking of something akin to stranger version of a Wimshurst machine for inducing high electric charges (and thus dumping energy) to nearby objects. Weird, not necessarily very stealth, but at least possible in principle.
Finally, the ultimate suit-internal heat sink is one for which I am aware of exactly zero plausible supporting pathways: Endothermic nuclear, the nuclear equivalent of those instant cold packs used in sports (please see @RobJeffries excellent comment below on that point). If you can imagine some perfect process that gathers all the  excess heat and "somehow" converts it without any major losses into the splitting of helium-3 nuclei into protons and deuterons, you have the idea.
A: I guess this is possible: for example, thermal radiation can be emitted within a very small solid angle, e.g., upwards.
A: Such technology is in its infancy, but it definitely exists. The images below are produced by several companies promoting their thermal/IR camouflage clothes. Obviously the applications are well-suited for the military, so who knows what more the military has developed.



This last image is made by a company called Blucher Systems. The link provides much more detail about the how and includes a neat little video of the suit in action. They claim a maximum of a 4 degree C difference between the "ghost" suit and ambient.
A: You couldn't make the heat "disappear". You can however make it go somewhere else.
To talk about thermodynamics very loosely
Anything (especially a vacuum) between the source and the detector will hide the source, as long as the medium remains at the surrounding temperature. This means that if you have a heat source behind a window, take a thermal imaging camera, and point it at the window, you will only see the temperature of the glass, not the temperature of anything behind it. 
However, if there is nothing for the window to transfer heat to, it will heat up (and the source will cool down) until it is the same temperature as the source.
While this is all fine and good for something you aren't in contact with, actually touching the glass will transfer body heat much faster, than if there was an insulator (such as air) between the two.
To make a suit to hide body heat, you would need a very insulated suit connected to a heat pump (refrigerator) with it's exhaust put out somewhere farther away, which would probably require insulated tubes too.
I can't see it being very practical for moving around in.
