If photons don't experience time, does that mean time is a consequence of mass? If photons don't experience the passage of time because they have no mass, does that mean time is a consequence of mass?
To me that's a profound conclusion yet not one I see printed. Is it right?
 A: I'm not sure where you're at in your studies, but you shouldn't accept anything as true just because it's in a book. You should convince yourself of it beyond any reasonable doubt. In the words of Boyle, "nothing by authority." 
This is related to this question, but I do not suspect the OP here will get much from the solution there, so I reword it more simply:
In special relativity, light rays (or just call them photons) are called "null observers." Null observers, by definition, have no proper way of measuring time intervals. Conceptually, one measures time intervals in relativity by sending light signals between frames - but if you are traveling at light speed then that measurement is useless. Mathematically, the proper time of an ideal clock between $t_{o}$ and $t_{1}$ is given by,
$$ \int_{t_{o}}^{t^{1}} \Bigg(1 - \frac{v^{2}}{c^{2}}\Bigg)^{1/2} dt = \int(0) = 0$$ 
since $v = c$ for null observers.
Perhaps this also helps for understanding "why null observers?"

If photons don't experience the passage of time because they have no mass, is it fair to say the property of time is a consequence of having mass?

I think it's more correct to say that they do not "pass time" because they are traveling at light speed, and they travel at light speed by definition. There is a very interesting article here about why photons have zero rest mass.
A: 
is it fair to say the property of time is a consequence of having mass?

Yes, it is fair to say that as long as you are specifically speaking of proper time (which seems reasonable to say when speaking of “the property of time”). 
Proper time is an affine parameter for timelike worldlines, and only massive objects have timelike worldlines. Massless objects do have affine parameters, but they are not proper time. 
A: It is true that photons do not experience proper time. On the other hand  a photon can experience a series of events during its lifetime. Does that means that he experiences all the events at the same time? this does not seem plausible. In addition, you cannot define a reference frame that travels at c. Mixing all of this together, one possible conclusion could be that the experience of proper time in a non-reference frame is different than the experience of proper time in a reference frame, and thus we have no clue what the photon "experiences".
A: I do not think it is fair to say the property of time is a consequence of having mass.
Photons, even though are not supposed to experience time per GR, but they somehow, still retain the frequency, which by definition involves time. 
Again, there has to be something massive to measure that frequency otherwise it does not mean much.
However, to say that universe looses track of time is going too far, because the frequency is remembered and is always kept available if there were anything to observe/measure it.
May be this part is off the topic, but IMO, universe always knows everything, including uncertainty and entanglement! Otherwise things at macro level would behave unpredictable. It is the process of observation that has limitations
A: 
To me that's a profound conclusion yet not one I see printed.

If there were just massless fields (particles), it would still be the case that spacetime has 1 temporal and 3 spatial dimensions so, in this sense, time exists independently of mass.
However, as Sir Roger Penrose has pointed out in his work on the Conformal Cyclic Cosmology (CCC), when there are just massless (conformally invariant) entities, there is no way to 'build a clock' (or a ruler) and so the universe 'loses track of time' (and distance too).
From Before the Big Bang: An Outrageous New Perspective and its Implications for Particle Physics 

Physically,  we may  think  that  again  in  the  very  remote 
  future,  the universe “forgets” time in the sense that there is no way
  to build  a  clock  with  just conformally  invariant  material.  This
  is  related  to  the  fact  that  massless  particles,  in relativity 
  theory,  do  not  experience  any  passage  of  time.   We might even
  say that to a massless particle, “eternity is no big deal”.

A: Not quite - but almost.
Time is, as mentioned before, a dimension of spacetime, so it - at least insofar as the ontology of special relativity is concerned - exists independently of objects which possess, or do not possess, mass to inhabit it or any of the other dimensions of spacetime. Thus it doesn't make sense to say that time per se is a consequence of mass.
However, there is something related to time that photons do not do, that things with mass do or at least can do: and that is evolution, the change of some aspects, properties or configuration of an object, system or organization with the marching-onwards of time. There are two basic types of evolution we can consider in relativity theory: one of these is internal evolution of the object itself, which is not only what it would "experience" but also which is in a real sense required for the concept of of "experience", which means in the most general sense to react or be affected internally to or by the goings on around oneself, to make any sense as applicable to that object. The other type of evolution is external evolution as seen by an outside observer - this corresponds to kinetic motion through space.
Where that mass enters in is that only particles with mass are capable of undergoing internal evolution, and thus, experience. Photons do not internally evolve, and thus not only do not "experience time", they do not experience, period, at least not in any way we could make sense of that physically and with our best understanding of things, i.e. the usual caveats that apply to our scientific knowledge. So while it is not correct to say that time is a consequence of mass, it could be legitimate to stake that internal evolution is a consequence of mass.
And the reason for this is that, fundamentally, mass is a form of energy, and what "energy" really is - instead of the various ideas that are often trotted like "energy is the capability to do useful work" (?! so if I have an engine sitting in a thermal equilibrium heat bath, then I have no energy, because I have no capability to do any work with this?) - is the generator of dynamism, that is, the thing that makes change, or evolution possible. Rest mass is the form of energy that makes the internal evolution above, possible, while kinetic energy (or "relativistic mass" - here it does make sense to invoke such a concept for we are comparing the two on a related level), generates external evolution, the evolution with regard to parameters that can only be defined by an external observer like its relative position.
The notion of energy as the generator of dynamism also comes up in quantum mechanics, where that it takes the form of a pseudo-operator that acts to shift a temporal sequence of wave functions back and forth, and through its equation to the Hamiltonian operator, then births the future history of the system (up to the informational limits imposed by the fact that $\hbar > 0$ which degrade the system description from a classical set of deterministic values to a spread of probable values of what can be extracted by an outside active agent which is what in quantum mechanics replaces and upgrades the passive observer of Newtonian mechanics and Einsteinian classical relativity) - this is what the Schrodinger equation is actually saying, and its true "meaning": the manner in which the energy is determined by the system's configuration at its present moment determines how it will become in the immediately succeeding moment (metaphorically, since of course there is no smallest increment on a real number time axis).
