The Standard Model has three fundamental forces, the electromagnetic force, the weak force, and the strong force. I outline the categories of particle interactions with these forces below, with the most plausible missing particle categories to explore in bold.
* Standard Model quarks which are fermions interact with all three of these forces. They have weak force interactions (in beta decay) mediated by W and Z bosons, they have strong force interactions mediated by gluons, and they have electromagnetic interactions mediated by photons. There are no known bosons that interact with all three of these forces (if there were, they would presumably be charged gluons with some rest mass and might be hard to distinguish experimentally from light charged mesons like charged pions and charged kaons, for example, perhaps either the short kaon or the long kaon is really a charged gluon with rest mass and not a composite particle).
* Charged leptons (which are fermions) and W+/W- bosons interact with two of these forces, the electromagnetic force and the weak force, but not with the strong force.
* Neutrinos (which are fermions), Z bosons and hypothetical Higgs bosons interact with one of these forces, the weak force, but not with the electromagnetic force or the strong force.
* Gluons (which are bosons) also only interact with one of these forces, the strong force, but not with the electromagnetic or the weak forces. No fermions are known that interact with the strong force, but do not interact with the weak force or the electromagnetic force. Such fermions would presumably have no charge and no rest mass, but would have a color charge.
* Photons (which are bosons) interact with one of these forces, the electromagnetic force, but not with the strong force or the weak force. No fermions are know that interact only with the electromagnetic force. Such fermions would presumably have charge, but no rest mass.
* Hypothetical gravitons (which are bosons) interact with none of these three forces. They have no charge, no rest mass, and color charge. There are no fermions that have no charge, no rest mass, and no color charge, although until recently the neutrinos were believed to be particles of this type and right handed neutrinos, if they existed would fit in this category. A scalar gravitational boson that interacted with no force but gravity could also have no rest mass, but without a force to carry, could be the prototypical dark energy carrier, making it a natural complement to the graviton as the implementation of the cosmological constant component of general relativity. Its absence of non-gravitational effects and constant speed of light movement would probably make the direct detection of such bosons impossible even if they did exist. This might be equivalent to the hypothetical graviscalar.
Three of the four gaps in these categories could be resolved if there was a law of physics that provided that there could be no massless fermions (a gap the might also explain the non-existence of right handed neutrinos, which would interact with nothing but gravity and might not even interact with gravity as rest mass seems to be linked to weak force interactions). There are several other conceivable combinations of interactions and fermion/boson status that have no particles associated with them.
* There are no fermions or bosons that interact with the two forces that are the weak force and the strong force, but not the electromagnetic force. The fermions would be electrically neutral quarks. The bosons would be gluons with rest mass.
* There are no fermions or bosons that interact with the two forces that are the electromagnetic force and the strong force. The fermions would be quarks with no rest mass, almost like up quarks except that nothing would ever decay into them. A no massless fermion rule would exclude them. The bosons would be electrically charged gluons.
Thus, to recap, the missing particle categories that would make sense if there was a no massless fermion rule would be as follows:
1. electrically neutral quarks
1. charged gluons with some rest mass
2. gluons with rest mass
3. electrically charged gluons
4. a scalar gravitational boson (perhaps associated with dark energy)
It is worth observing that there are composite analogs to three of the four strong force interacting missing categories.
* There aren't any electrically neutral quarks (basically neutrinos with color charge), but there are quite a few electrically neutral baryons that are fermions, most famously, the neutron.
Electrically neutral quarks, if they existed, might be good dark matter candidates, since their stable hadrons might be lighter than ordinary baryonic matter for deep reasons that somehow match the reality that neutrinos are lighter than charged leptons. Since they would have mass, they would have weak force interactions, at least through Z bosons. To have W+ or W- interactions, there would have to be quarks with an electrical charge of +/- 1 as well (basically electrons with color charge), to mimic neutrino-charged lepton couplings. But, their absence from the Z boson decay spectrum argues against their existence.
* There are electrically charged mesons (e.g. charged pions and charged kaons) that are electrically charged bosons with rest mass that have color charges that can lead to strong force interactions within them.
* There are electrically neutral mesons (e.g. neutral pions and neutral kaons) that are bosons and have rest mass that have color charges that can lead to strong force interactions within them. Glueballs are also hypothetical bosons that have rest mass and color charges that can lead to strong force interactions within them, even though isolated gluons lack mass.
One could imagine that Nature lacks fundamental particles that simply do what composite particles could out of some sort of cosmic avoidance of superfluous particles rule. Alternately, one could imagine that some of the so called fundamental particles are themselves composite particles made out of preons, so that all of the available categories (or at least many of them) are actually filled by composite particles.
There are no electrically charged gluons without rest mass. Indeed, there are no electrically charged particles of any kind without rest mass; all massless particles lack electromagnetic charge. Perhaps, as a consequence of the deep links associated with electroweak unification and the apparent role of the weak force in giving rise to fundamental particle mass (since all massive particles have weak force interactions and no massless particles have weak force interactions), it is impossible for charged particles to lack rest mass entirely.
The converse proposition, that massive particles must have electromagnetic charges, is not true, unless neutrinos and Z bosons and Higgs bosons are actually all composite particles (a not completely absurd possibility).
The Higgs boson mass is remarkably close to the sum of the W+, W- and Z boson masses divided by two (or perhaps more generally, the sum of all four electroweak boson masses including the massless photon), suggesting some sort of linear combination of them, which has a composite character to it.
It is not hard to imagine a scenario in which the Z boson is a composite of the W+ and W- bosons although it doesn't fit the story of electroweak unification very well, indeed, electroweak unification imagines the Z boson and photons as non-trival linear combinations of a B boson and a neutral W boson via Higgs boson interactions, which makes both Z bosons and photons, in some sense, quasi-composite even in the Standard Model.
Composite neutrinos are a harder pill to swallow, but the neutron certainly provides an analogy, and if light particles can turn energy into mass to emit W and Z bosons, while W and Z bosons can turn mass into energy to emit particles lighter than themselves (or at higher energies, particles heavier than themselves), it certainly doesn't defy nature to imagine that composite neutrinos could have masses less than their component parts.
This isn't to say that any of these particles are actually composite, although they might all be varying manifestations of a single fundamental particle in different configurations a la string theory. The notions of democratic Z boson decay and quark-lepton complementarity, for example, discourages the idea of thinking about any fermions as composite. They all seem to be on a equal footing (or at least 3-1 proportionality due to color charge distinctions) in some sense.