* There is only one force that interacts only with particles with rest mass (all fermions and W bosons and Z bosons) and does not interact with particles without rest mass (photons and gluons). It is the weak force. Of course, one could argue that it only interacts with left handed and not right handed versions of those particles. But, still, there is not entire class of particles which does not have weak force interactions.
The weak force is also the only fundamental force that changes the rest mass of a particle thought its interactions.
The weak force causes particles with greater rest mass to decay into particles with lower rest mass more often than it gives rise to particles with greater rest mass from lower rest mass (of course, in all cases subject to total mass-energy conservation). And, in general, the greater a particle's rest mass, the more rapidly is decays.
The mass of the W and Z bosons was accurately predicted from theory using a unified electroweak theory and empirically proven physical constants that do not involve the strong force or gravity.
* The strong force does not interact with leptons, W bosons and Z bosons which have rest mass. It does interact with gluons which lack rest mass.
The strong force appears to play a significant role in the mass of composite particles that have color force charge, as illustrated, for example, by models of the mass of glueball in low energy QCD. I don't understand QCD well enough to know if the role of the strong force in establishing mass of strong force interacting particles is entirely a product of binding energy or has some additional component as well. For example, about 90% of the mass in the nucleons is due to the strong interaction between quarks and gluons, and the mass of the quarks and/or EM interaction between the quarks contribute the remaining [mass] according to QCD calculations of proton mass from first principles. (Spelled out in greater detail here and here.) Indeed, since the vast majority of the mass in the universe that is neither dark matter nor dark energy comes from protons and neutrons, and 90% of the mass in protons and neutrons comes from strong interactions, strong interactions account for the vast amount of ordinary matter rest mass in the universe.
Quarks, which have color charge, are much heavier than leptons, although the W and Z bosons are quite heavy in the absence of color charge.
* It is true that both of the particles that lack rest mass (the photon and gluon) also lack electromagnetic charge. But, the converse is not true. Neither the massive neutrinos nor the Z boson have an electromagnetic charge, but each of them do have a rest mass.
The electromagentic charge neutral baryon called the neutron, and the electromagnetic charge neutral Z boson, are both heavier than their charged counterparts the proton and the W boson.
Also, while the photon does not have an electromagnetic charge of its own, it does participate in electromagnetic interactions as its carrier boson, despite its lack of rest mass.
* Gravity influences the course taken by photons that have no rest mass as well as particles that do have rest mass. Conversely, energy as well as rest mass can induce gravity in the stress-energy tensor.
More speculative points linking the weak force and mass
* The Standard Model Higgs is really a Higgs triplet. But, two of the members of the Higgs triplet are "eaten" by the W and Z bosons which are the carrier bosons of the weak force. Given the central role that the hypothetical Higgs boson plays in the creation of inertia mass, this is an interesting fact.
* The Higgs boson, which is presumed to be quite massive, is also presumed to experience weak force decay that can be observed. It is also presumed to lack spin and electromagnetic charge and color charge.
* Composite particles in which a significant share of the particle's rest mass is attributable to binding energy appear to behave no differently in terms of inertial or gravitational mass than fundamental particles of the same mass.
This is suggestive of the idea of rest mass as a quantity that could be the product of multiple kinds of interactions.
It is also suggestive of the idea that mass could be generated mostly, or entirely, by the binding interactions between massless or extremely light preons, or alternatively, from weak force interactions between fundamental particles.
* There is considerable intuition among particle physics researchers that there is some deep relationship between the masses of the fermions and the entries in the CKM and PMNS matrixes. It seems likely that a single underlying principle, if there is one, explains both of these sets of constants in the Standard Model from some more fundamental theory. The relationship seems to be one in some way linked through the square root of rest mass rather than rest mass itself. We normally think about mass as driving transition matrixes, but we could also imagine transition matrixes driving masses. Indeed, if the two are bound up in some master equation, asking which one drives which may be a nonsense question.
* Z boson interactions do not themselves appear to produce changes in the rest mass of the particles that emit or absorb Z bosons, although Z boson decay does involve products that differ in rest mass from the rest mass of the Z boson.
* One of the predominant weak force interactions in nature is beta decay (the other is neutrino oscillation). In its most typical form it involves a bottom quark emitting a W- boson and becoming an up quark. The W- boson in the typical case (and sometimes the only case permitted by mass-energy conservation) then typically, in turn, decays into an electron and anti-neutrino of the electron neutrino type. The rest mass of the bottom quark is not precisely the same as the rest mass of the up quark plus the rest mass of the electron plus the rest mass of the electron anti-neutrino, but the mass gap between the source and the products in beta decay is very slight.
The rest mass gap in weak force interactions higher generation particles of the same types of fermions is much greater.
* Rest mass is not the only factor at work in the rate of weak force decay of composite particles. Their decay rates are also related to spin (with higher spin particles decaying more rapidly) and perhaps to electromagnetic charge also (with higher charge particles decaying more rapidly).
* CP violations in the weak force are associated with high rest mass fundamental particles that participate in the weak force interaction.
* While we think of the weak force as being weaker than the strong force, weak force driven top quark decay involving the highest mass fundamental particle happens before the strong force has an opportunity to hadronize the top quark, even though the weak force boson is massive and hence slower than the massless gluon which presumably moves at the speed of light.
* All particles with rest mass except up and down quarks in protons and neutrons that are bound into atomic nuclei, and electrons, are unstable. Even electron neutrinos and electron anti-neutrinos oscillate rather than maintaining a stable ground state.
* The theorized graviton, would have spin-2, no rest mass, no electrical charge and no color charge. If the rest mass/weak force hypothesis holds true, this boson would not have weak force interactions either.
* All quarks and charged leptons are believed to have "Dirac mass" related to Higgs field interactions, while there is some speculation that neutrinos might derive their mass from anthor source called "Majorana mass" or from a combination of the two kinds of mass. See, e.g. here. The non-detection of neutrinoless double beta decay to date, however, subject to increasingly stringent experimental limits, casts doubt on the existence of Majorana mass as a physical proces.
* We infer neutrino mass from its weak force oscillations, not necessarily from its direct detection.