How does general relativity affect a gravitational acceleration problem?

Say you create two portals, one above the other, so that an object falling from the higher portal will land in the lower portal, creating a loop. Ignoring terminal velocity (possibly taking the problem to space), how do relativistic effects alter the problem of an infinitely falling object compared to Newtonian physics? At what point do we start worrying about the object pulling Earth around? And how much speed must a one-kilogram rest-mass companion cube acquire in order to vaporize a cube 3 meters on a side of either silicon dioxide or concrete?