Anything travelling faster than light would not just be an interesting challenge to General Relativity, it would mean that the causality principle is wrong. I would hate seeing this happen as this has been considered perhaps the strongest principle of physics ever and many interesting ideas have been thrown away just because they would be incompatible with perfect causality somehow.
Broken causality, together with an assumption of the existence of some form of free will, could mean, e.g., being able to drain energy which resulted from some process before it happens, and then prevent it. Repeat this an unlimited number of times and you have a perpetuum mobile. That's just one of the reasons why I am most suspicious about this result.
I have read the experimental paper briefly and I am attaching it in the Sources below for others to see. I suspect that the problem lies within the measurement of the neutrinos' passage: these particles interact weakly enough to pass 730 km of solid rock as if it was free space, so one can't expect them to generate a click in detectors at both ends of their journey. If I understand it right, they timestamp the emission of a neutrino indirectly by observing a muon that originated simultaneously with its production. However, this simultaneity is predicted only by the Standard Model, which is already well-known to be imperfect, *especially* in the neutrino section. If there was anything that delayed the production of a muon by as little as 60 nanoseconds, e.g., any kind of intermediate particle, the mystery is explained.
I'd welcome any comments on my opinion by real particle physicists.