Not necessarily. (Disclaimer: Physics background but this is not my area of expertise; I am working from memory of courses I took >5 years ago). In electroweak unification, there are four underlying gauge fields, superpositions of which make up the photon, W bosons, and Z boson. You have to adjust the coefficients of the combinations very carefully to make the photon massless and the weak bosons heavy. You could adjust them slightly less carefully and have an extremely light, but not massless, photon, without touching the underlying gauge fields; then you can derive Maxwell and whatnot using the gauge fields instead of the physical particles, and presumably save SR as well.
Observe that the current experimental upper limit on the photon mass (well, I say current - I mean, the first result that comes up in Google; it's from 2003, but not many people bother with experimental bounds on this sort of thing) is 7x10^{-19} eV, or what we call in teknikal fiziks jargon "ridiculously tiny".
SR doesn't depend on behaviour of gauge fields. Special relativity is necessary to have a meaningful definition of "particle" in field theory. The gauge fields have to have zero mass term because of gauge invariance, not Lorentz covariance. The mass is generated by interaction with Higgs particle, this is essentially a trick which lets you forget gauge invariance after the model is postulated. It doesn't impose any requirements on SR either.
http://www.nature.com/news/2011/110922/full/news.2011.554.html
http://arxiv.org/abs/1109.4897v1
http://usersguidetotheuniverse.com/?p=2169
http://news.ycombinator.com/item?id=3027056
Perhaps the end of the era of the light cone and beginning of the era of the neutrino cone? I'd be curious to see your probability estimates for whether this theory pans out. Or other crackpot hypotheses to explain the results.