Whoever the PRL referee(s) was/were, they should have sent it back to the author to put the argument into a manifestly covariant formalism. The editors should have done the same before the paper got to a referee. As it is, everybody has to waste time unpicking the 3-d vector mess. 3-d vectors have a perfectly legitimate place in Physics, but not if one is constructing arguments concerning Lorentz invariance/covariance or otherwise.
The title is misleading on the face of it, because the problem is reported as a failure of the system to conserve momentum, which is associated with translation invariance, not with Lorentz invariance.
There is only a "problem" if one is using the macroscopic form of Maxwell's equations. If one is using the macroscopic equations, the system will not be translation invariant if the material background is not homogeneous, similarly for rotation invariance and isotropy. If the background material is not homogeneous (and isotropic), momentum (and angular momentum) will not be a conserved quantities. Introducing the Einstein-Laub formula as a way to jury rig a non-covariant formalism is significantly too ad-hoc.
In any case, if a manifestly Lorentz and translation invariant Lagrangian can be constructed for a model, the forces that act in that model can be presented in a manifestly covariant way. The force equations could be arbitrarily complex, depending on what Lagrangian we introduce. The Einstein-Laub force law can only be applicable in a restricted setting, just as the Lorentz law is. One comment on the Science article linked to in comments points to a more-or-less intuitive resolution, "So what is wrong with polarization and magnetization being fundamental, given that point particles carry angular momentum and the quantum vacuum can be polarized?" Ultimately, this would have to be cashed out with a Lorentz and translation invariant Lagrangian (and then it would have to be quantized, etc.), but this seems the most positive thing to take from the paper. A panoply of Lorentz and translation invariant equations could be written down that include
the displacement and the magnetic induction as well as the electric field and the magnetic field as dynamical degrees of freedom, though proving anything about any given system might be prohibitively difficult.
A lot more could be said, and I have a feeling more will be said because the paper has been linked to across the web, by ZapperZ, for example, on May 3rd. Now that the paper has been published, it's fair game. It's different enough from what most people are doing in Mathematical Physics, however, that relatively few people will care much. Anyone who is busy with their own research is unlikely to comment, unless, like me, they're cross enough. I've now commented on this paper twice (at ZapperZ long ago), however, so it's time to join the ranks of people who are ignoring it.
EDIT(5/24/2012): ZapperZ has added two postscripts about new rebuttals on arXiv.