Massless charged particles

There's no problem in writing down a theory that contains massless charged particles. Simple $\mathcal{L} = \partial_{\mu} \phi \partial^{\mu} \phi^*$ for a complex field $\phi$ will do the job. You might run into problems with renormalization but I don't want to get into that here (mostly because there are better people here who can fill in the details if necessary).

Disregarding theory, those particles would be easy to observe assuming their high enough density. Also, as you probably know, particles in Standard Model compulsorily decay (sooner or later) into lighter particles as long as conservation laws (such as electric charge conservation law) are satisfied. So assuming massless charged particles exist would immediately make all the charged matter (in particular electrons) unstable unless those new particles differed in some other quantum numbers.

Now, if you didn't mention electric charge in particular, the answer would be simpler as we have massless (color-)charged gluons in our models. So it's definitely nothing strange to consider massless charged particles. It's up to you whether you consider electric charge more important than color charge.

Another take on this issue is that Standard Model particles (and in particular charged ones) were massless before electrosymmetric breaking (at least disregarding other mechanisms of mass generation). So at some point in the past, this was actually quite common.

Massless charged particles can't exist in Nature because they would be easily produced by the colliders, and they haven't been. Such a production would simply arise from the Feynman diagram with an intermediate photon that "splits" into the new charged massless particle and its antiparticle. The cross section of this process would be calculable, and not small in any way.

Also, the fine-structure constant $\alpha=1/137.036$, one expressing the strength of the electrostatic interactions in the natural units, is not a real constant. It's running. However, it's only running at energy scales such that there exist lighter charged particles. In Nature, it means that the constant is only running above the mass of the electron or positron - the lightest charged particles.

If there were massless charged particles, the electron and positron would become unstable - one problem - and the fine-structure constant would run to $\alpha=0$ at very long distances - another problem, and it obviously doesn't. So massless charged particles are theoretically impossible in our world - assuming that we empirically know some things such as the fact that there is a limiting Coulomb force at long distances.

Lubos' answer is good for the present day universe. Here I reply to the question made duplicate "Can a massless object have a charge associated with it? [duplicate]".

In the standard model of particle physics, before symmetry breaking in the cosmological model, all symmetries exist and all the particles with their quantum numbers exist. The gauge bosons are zero mass before symmetry breaking, and also all the fermions were zero mass before. So if we include the time of the universe in the question the answer is yes. Experiments have not reached the energies and conditions necessary to reproduce the conditions before symmetry breaking, so the answer for present day particles is given by Lubos' answer.