It’s extremely, EXTREMELY doubtful that e=mc^2 is not accurate. Even if the neutrinos really are going faster than light!
What would it mean? That depends on what neutrinos are. They are one of the most mysterious particles in our particle bestiary. For a long time, physicists thought neutrinos had no mass (and so were sort of like particles of light). Then they found out something really weird about neutrinos: they spontaneously change between three different “flavors”—electron, muon, and tau.
In order for neutrinos to “flavor oscillate,” they should have mass. But they still travel close to the speed of light. So their mass must be really small. Or at least that’s what physicists thought! As you can see from the equation, mass is closely related to light speed. And what’s really weird (to me, at least—I could be getting this wrong) the three neutrinos have different masses! So a neutrino seems to be able to spontaneously change its mass between three figures. (Although we don’t actually know if the different kinds have different masses). This flavor oscillation is apparently a quantum mechanical effect so I’m not going to pretend to understand what little we do understand about it.
If neutrinos really are traveling faster than light, I think the most novel change we’ll see will be a better understanding of just WTF neutrinos are and how they work. But as many people have pointed out, it’s extremely premature to conclude that they even are traveling faster than light. It could well be a measurement error.
Note: come to think of it, we still don’t even understand how mass works, or what it is, exactly. That’s what all the fuss over the Higgs boson is about. It seems to me like these two mysteries might be related.