Like others have said, any movement will propagate at the speed of sound through the rod. But moreover, as the end of the rod speeds up, it will experience relativistic mass increase from your point of view, so it would still take an infinite amount of energy to accelerate it to c, regardless of how long your lever was. (Though special relativity, dealing with inertial frames, is strictly not applicable to a rotating rod, but you can probably make use of a succession of infinitesimal inertial frames to figure that out.)
Good point. An unobtainium rod as JanezStupar calls it, would be the immovable object, stuck in the universe's gullet gumming up the works. I wonder if God would be able to move it after he created it.
