The tried and true reason to refrain panic is that everything is preliminary. These are anomalous results of an experiment, and nothing more. Other scientists have to verify the results, other scientists have to duplicate the effect, and only then can everyone panic.
Even then, there is no reason to panic for one simple reason: Einstein's theory of relativity never actually said that particles can't go faster than the speed of light. Rather, the mathematics dealt with the behavior of bodies as they approached the speed of light. And even then, it dealt only with the observable behavior of bodies approaching the speed of light.
The notion that nothing can go faster than the speed of light is an inferred result of the theory, not an implied one. The theory only implies that we can't observe anything accelerate past the speed of light.
What happens if it's verified that neutrinos do indeed travel faster than the speed of light? It's hard to predict. The second postulate of relativity (the speed of light is a fundamental constant, independent of the observer) would have to be revisited and assessed again. But perhaps they will find that the speed of a neutrino is itself a fundamental constant, and we have an adjusted postulate that abandons the speed of light altogether.
But, more likely, nothing is going to come of this. Anyone remember Cold Fusion? Anyone remember how the world of physics was turned on its ear when researches in Utah announced fusion at room temperature? Everyone was doing backflips over the implications, right up until no one else could duplicate the results, and the experiment turned out to be a flawed ball of cattle crap.