At the risk of being a bit redundant, I believe everyone is making too much of the the danger theoretical physics faces should the FTL neutrino data hold up.
As far as relativity is concerned, it would not affect the speed of light, at all. That is a measured value.
Special relativity could require some addition of an exception to the rule, but has proven locally accurate at all other tested scales. Though to some extent the a priori facts SR is based on, could be seen as challenged, the general underlying conceptual model has been well supported by existing experience and would not change.
General relativity would be affected not at all. Besides it is already coming under some strain and need of at least clarification if not some reconstructive modernization, based on galactic orbital velocities and the whole dark matter discussion.
Quantum mechanics would probably require the most attention, but I am sure tweaking the math is not beyond the skills of those directly involved in the field.
As I mentioned earlier, a FTL neutrino.., with mass, would probably be a bigger challenge to our understanding of inertia and through the equivalence principal gravity, at least with respect to our fundamental understanding. Not so much in a complete over haul, as in establishing some limitation(s) of scale. We already have the gravitational conflict involving GR and QM, which at present we just sort of look past or ignore.
If confirmed this could actually open some doors to a better understanding of both inertia and gravity. It would in some respects force the issue. Right now what stand in the path of the FTL limitations on mass, is inertia. If inertia itself were to have a fine structure scale or texture, it would go a long way toward explaining the difficulty GR has modeling gravitation at subatomic scales, explaining how a neutrino could slip through the cracks and exceed the speed of light, and provide new opportunities to develop a working model of quantum gravity. The last of which would be a welcome step toward a truly united theory.
Note: While Einstein's field equations model gravity as a curvature of space resulting from an interaction of matter/mass and space, they do not explain how space and matter/mass interact to accomplish this, just that they do. It would be nice to know how. GR models space as smooth, while QM gives it a texture. If space really has a texture and the smoothness modeled by GR is only an approximation, again doors open between QM and GR, as well as holes in inertia that the neutrino might just slip through...
Disclaimer: This does not represent a prediction. It is mealy looking at the current situation with an eye toward some of the unanswered issues we already have on the books.
I think most of the initial reactions by recognized physicists, that seem concerned is an automatic visceral reaction to something that at first glance appears to challenge their world paradigms. When the dust settles most of that initial reaction will dissipate and if the data holds up there will be flurry of new research proposals, competing for the growing limits in research funding.