Is (rest) mass quantized?

There are a couple different meanings of the word that you should be aware of:

• In popular usage, "quantized" means that something only ever occurs in integer multiples of a certain unit, or a sum of integer multiples of a few units, usually because you have an integer number of objects each of which carries that unit. This is the sense in which charge is quantized.
• In technical usage, "quantized" means being limited to certain discrete values, namely the eigenvalues of an operator, although those discrete values will not necessarily be multiples of a certain unit.

As far as we know, mass is not quantized in either of these ways... mostly. But let's leave that aside for a moment.

For fundamental particles (those which are not known to be composite), we have tabulated the masses, and they are clearly not multiples of a single unit. So that rules out the first meaning of quantization. As for the second, there is no known operator whose eigenvalues correspond to (or even are proportional to) the masses of the fundamental particles. Many physicists suspect that such an operator exists and that we will find it someday, but so far there is no evidence for it, and in fact there is basically no concrete evidence that the masses of the fundamental particles have any particular significance. This is why I would not say that mass is quantized.

When you consider composite particles, though, things get a little trickier. Much of their mass comes from the kinetic energy and binding energy of the constituents, not from the masses of the constituents themselves. For instance, only a small part of the mass of the proton comes from the masses of its quarks. Most of the proton's mass is actually the kinetic energy of the quarks and gluons. These particles are moving around inside the proton even when the proton itself is at rest, so their energy of motion contributes to the rest mass of the proton. There is also a contribution from the potential energy that all the constituents of the proton have by virtue of being subject to the strong force. This contribution, the binding energy, is actually negative.

When you put together the mass energy of the quarks, the kinetic energy, and the binding energy, you get the total energy of what we call a "bound system of $\text{uud}$ quarks." Why not just call it a proton? Well, there is actually a particle exactly like the proton but with a higher mass, the delta baryon $\Delta^+$. Technically, a $\text{uud}$ bound system could be either a proton or a delta baryon. But we've observed that when you put these three quarks together, you only ever get $\mathrm{p}^+$ (with a mass of $938\ \mathrm{MeV/c^2}$) or $\Delta^+$ (with a mass of $1232\ \mathrm{MeV/c^2}$). You can't get any old mass you want. This is a very strong indication that the mass of a $\text{uud}$ bound state is quantized in the second sense. Now, the calculations involved are very complicated, so I'm not sure if the operator which produces these two masses as eigenvalues can be derived in detail, but there's basically no doubt that it does exist.

You can take other combinations of quarks, or even include leptons and other particles, and do the same thing with them - that is, given any particular combination of fundamental particles, you can make some number of composite particles a.k.a. bound states, and the masses of those particles will be quantized given what you're starting from. But in general, if you start without assuming the masses of the fundamental particles, we don't know that mass is quantized at all.

I want to offer a different perspective from the already existing answers, which all seem to somehow refer to the Standard Model or other specific physical theories to say that mass is not an integral multiple of some fundamental mass unit, hence not discretized. The reason why mass is not like that - and can indeed conceivably have continuous values in a consistent quantum field theory - is inherently related to the properties of the symmetry it is the "charge" of: Poincaré invariance.

All symmetry groups in quantum physics must be represented by a projective unitary representation on the Hilbert space of states (cf. Wigner's theorem). It is this fact alone that forces discretization of many quantities.

The archetypical example of a discrete quantity in quantum physics is spin, coming in integral multiples of $\frac{1}{2}$. Spin is discrete because it is the value of the quadratic Casimir operator $S^2$ of the rotation group $\mathrm{SO}(3)$, which is constant on its irreducible representations, and the rotation group has only countably many irreducible unitary representations, since as a compact group, all its irreducible representations are finite-dimensional, and there are only countably many finite-dimensional vector spaces. Furthermore, it turns out that, purely representation-theoretic, the only existent irreducible representations are those where $S^2$ takes values as $\frac{n}{2},n\in\mathbb{N}$.

Likewise, mass is the (square root of the value of) the Casimir operator $P^2$ of the Poincaré group. The Poincaré group is non-compact, which means it does not have finite-dimensional unitary representations. Therefore there is no reason to expect there to be only countably many of them, and, in fact, there are not. By Wigner's classification, there is a unitary representation (several in fact, labeled by the spin of the massive particle) of the Poincaré group for every possible positive real value $P^2 = m^2\in\mathbb{R}_{>0}$. Therefore, there is no reason that the usual procedure of quantization (which is not "associating discrete values to things" but rather something like "representing all physical observables and symmetries as operators on a Hilbert space") should confine masses to a discrete spectrum, let alone one where all masses are integral multiples of a fundamental mass unit. Physically only finitely many of the representations will be realized because we only have a finite number of distinct particle species, but, in contrast to spin, there is no a priori constraint of the masses in a quantum theory.

I think just the opposite of David Zaslavski, and assert:

The rest mass of particles is quantized, [edit] being the spectrum of the component P_0 of 4-momentum in the Hilbert space of states where the particle is at rest. (For example, quarks and neutrinos have a 3-dimensional mass matrix, each eignevalue being infinitely degenerate.)

This doesn't conflict with David's supporting facts but with his use of the terminology. For:

(i) A quantity is conventionally called quantized if its spectrum (the set of possible values it can attain) is discrete. This is the case for mass, as mass is defined in quantum field theory as values of the energy where the S-matrix in the rest frame becomes singular (''poles of the S-matrix''). Such poles must be discrete in each instance, for purely mathematical reasons. More specifically, the masses of the known elementary (and less elementary) particles are tabulated and can be seen to take fixed, discrete values.

(ii) Being quantized has nothing to do with being a parameter. Indeed, electromagnetic carge is quantized, although the value of the electric charge is a free parameter of the standard model.

(iii) Being quantized has nothing to do with having a (simply or not) understandable pattern. Most spectra of chemical substances have know pattern, but they are all expained by the discreteness of the spectrum of the corresponding Hamiltonian - the most conspicuous case of quantization.