"So, just what would happen if they unexpectedly push a proton faster than light? "
Nothing. Here follows a *very* simple explanation:
Why can a plane go faster than sound? Because the plane is not made out of sound-waves but of matter (light-waves).
Why can't we go faster than light? Because all matter is (if you'll excuse the simplification) made of photons... and you can't push a photon to travel faster than light, since it itself *is* light.
Bottom line, if you do want to travel faster than light you would either have to:
1) somehow change matter/energy into something completely different, then re-integrate it at the destination (effective teleportation; and no, the Casimir Effect won't help you much);
2) "drop out" of our universe into one where "distances" are smaller, then "drop back" into our universe (so-called "hyperspace travel"); and/or
3) force the universe to "wrinkle" to allow you to "shrink" the distance between you and your destination (space-warps or wormholes).
My apologies to any *real* physicists who will be reading this - I fully acknowldge that this explanation has been at Pterry's "Lies To Children" level. Those interested in a more complete reasoning can do their own research. ^_^