What fundamental principles or theories are required by modern physics? We have been taught that speed of light is insurmountable but as we know an experiment recently tried to show otherwise. 
If the experiment did turn out to be correct and confirmed by others, would it make physics to be rethought of? What other concepts are fundamental to physics, which, if disproved would need radical rethinking?
If this sounds too juvenile and/or misinformed, please understand that I am a layman, having nothing, professionally or academically to deal with science, directly, and this question is out of curiosity. I have developed a liking to "science stuff" and been reading popular science variety of literature lately. This question was also prompted by what Sheldon Cooper had to say in one of the episodes (I was watching a rerun).
 A: A similar foundational cornerstone of physics is the principle of rotational invariance. Suppose that the laboratory finds that neutrinos (or anything else) have different oscillation rates when going in the N-S direction than in the E-W directions, in a vacuum, with no relation to anything else. This would break physics just as badly as faster-than-light neutrinos. If a laboratory announced this result, they would be laughed at, but the faster than light neutrinos are the same thing with respect to space-time.
There are no published tests of rotational invariance which are as good as the tests of relativity, partly because rotational violation is counterintuitive so no-one bothers. But for a modern physicist, relativity violations are counterintuitive in the exact same way.
A second foundational cornerstone is translation invariance. This is the principle that there is no way to tell where you are in an absolute sense, without measuring relative to something else that's there. If we found a magic spot--- a position where muons didn't decay for example, and this spot was just somewhere, you couldn't get it to move, this would be a violation of translation invariance. Translation invariance is even more fundamental than rotational invariance.
The experiments which would show violations of these are:


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*violations of momentum conservation law

*violation of energy conservation

*violation of angular momentum conservation

*violations of the law of center-of-mass motion.

*violations of CPT (matter/antimatter symmetry)


If the Neutrino observations hold, they lead to a violation of center-of-mass conservation law for sure. You can move something's center of mass in one direction without emitting anything, just by propagating superluminal neutrinos one way, converting them to photons, and sending the photons back the other way.
Other than the space-time symmetries, the other inviolable basic principle is quantum mechanics. If you find a particle whose position and momentum are not uncertain, or which is not described by probability amplitudes, then you break quantum mechanics. This is difficult to imagine, because if one part of the world can be superposed, it is difficult to see how another part doesn't get the superposition by interacting with the first part. But the principles of quantum mechanics allow a deformation with decoherence, and this gives the Lindblatt formalism for density matrices. So a violation of Quantum mechanics is usually thought of as a certain amount of irreversible decoherence


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*No irreversible decoherence in fundamental systems


These are the main experimental facts on which modern physics is built which could not be accomodated easily by modern physics. The first four are pretty sure, but Hawking tried to get irreversible decoherence in black hole physics as recently as 10 years ago.
If you put these together, there are relatively plausibile deductive paths that lead to relativistic field theories that are used today. If you add some assumptions both of a gravitational and non-gravitational nature, you find string theory should be the correct gravity theory. There is nothing below string theory, so you are done in terms of fundamental theories.
A: Not very much !
The rest of physics still has to work, finding that the speed of light can be exceeded in certain circumstances doesn't suddenly change the results in other experiments or allow perpetual motion machines to start working. 
There have been some discoveries where things that were classically 'impossible' were found to work in quantum theory - which have led to practical discoveries (like SQUIDS or even GMR hard drives). Although it's hard to see how you could practically speed up the internet using oscillating neutrinos.
edit: The actual Opera experiment looks like a mistake.
But imagine if it was discovered that (for example) you could send a signal faster than light by some QM effect - but over a distance <0.1nm and only below a temperature of 1mK that would invalidate relativity but have no affect on the day-day use of relativity in physics or on the structure of the universe.
In exactly the same way that a tiny difference in the orbit of mercury overturned Newtonian mechanics and led to GR but  had no effect on the day-day use of Newtonian mechanics for calculating the flight of cannon-balls!
A: Most progress in physics is incremental to start with, as far as data and experiments go. Theories change following new data but on the whole they change by incorporating the old theories as limiting cases for certain parameters of the new theories, or convolutions over the variables of the new theories.
@Ronmaimon's list is valid , and if an experiment violates one of these conditions the theories would have to be reshuffled/reformulated or, as has happened in the past, the phenomenon explained by new particles. I remind that the neutrino was discovered because energy and momentum conservation had to hold, for example.
The Standard Model of particle physics has to be incorporated in any new theory because it is a shorthand for all the data up to now with very few dark spots ( CP violation comes to mind). Incorporated does not preclude new ways of looking at the data, just that there should be consistency with the old.
If strings are the theory of everything, on the other hand they bring us many unexplored dimensions, and if we have managed to have such complicated theories with 3+1 dimensions,  God knows what smart theorists can come up with trying to accommodate violations, and they are already exploring theories to  fit  these superluminal neutrinos if they turn not to be a systematic error.
A: Part of the problem in confirming physics theories with experiments is that we don't know all of it. Most likely, there are unthought of circumstances and margins of error in the "neutrino experiments". If discrepancies occur in experiments of such well tested theories, more rigorous testing remains before failure of the theory is considered. Not to mention that any subsequently reworked theory has to at least satisfy the old theory as well.
So, yes - given sufficient evidence to topple a theory, physicists would go through the five stages, and eventually have no other choice but to yield - science is about the world, not the ego.
