Terrazzo is a pretty hardy floor so the carpet issue should be okay if it was only glued down. You might have residue that would need to be removed. If tackless was nailed into the floor, it would need to be repaired.
What catagory of tile are you talking about? Vinyl or ceramic? The least problematic would be the self stick variety. The others would have residue to deal with.
Best thing to do would be to lift up a corner of the carpet and see what you had.