Wouldn't this also violate conservation of energy? A particle with any mass, traveling at the speed of light, would have infinite energy. And accelerating a particle to that speed would "push back" on the earth with infinite force, so it might also break conservation of momentum.
All the laws of physics are closely interlinked. Generally, an event or situation that violates one law of physics must also violate many others as well. For example, instantaneous teleportation would violate conservation of energy (because the departure and arrival points would have different gravitational potentials), relativity (because of faster-than-light travel), and conservation of momentum (because you need to adjust your velocity to match the arrival point in order to arrive safely), among others. This doesn't mean that teleportation violates the laws of physics in three separate ways. The three violations are all different ways of looking at the same physical impossibility.
The reason a particle would have infinite energy as it approaches the speed of light is because its mass (relativistic mass) would also approach infinite. Thus the understood reason a particle could not attain the speed of light is because most of the energy added into a particle going near the speed of light turns into relativistic mass instead of velocity.
If these neutrinos are indeed going faster than light, they clearly did not become infinitely massive.
The concept behind tachyons is that whilst particles can't approach the speed of light they can be faster than the speed of light and not break relativistic equations (they couldn't then slow to the speed of light either). Similarly with time reversal.
Personally I think this is cool but probably also just difficult (if not impossible) to falsify and so hasn't yet been restricted. When you apply a mathematical model it's like applying a metaphor [so meta!], it has a limit at which it breaks down but that limit isn't always obvious and sometimes at the edges things look tantalisingly like they might work.
Relativity does not permit particles with mass to go the speed of light. However, the math works just fine with massive particles going faster than light. Further however, the effects are quite bizarre, even if well defined. You know how your physical intuition is basically useless when it comes to understanding relativity? It's even more useless when understanding what the math says about tachyons.
Just to be clear, I did say "approachable". I've seen heavier duty explorations of what tachyons are like under real relativity theory, but I don't know of anything else that approachable.
