Tear off a section of aluminum foil that is large enough to cover the damaged tile. Lay it over the tile. Plug in a hair dryer and set it on hot. Aim it at the tile for five to 10 minutes to loosen the glue holding the tile to the floor.
Hold the putty knife so that the blade is parallel to the floor. Work the edge of the knife under the edges of the damaged tile at a 45-degree angle. Pry the scratched plastic tile off of the floor, heating it with the hair dryer as needed in order to get the tile off of the floor in one piece. Discard the scratched plastic tile.
Pour a quarter-sized amount of mineral spirits onto a rag. Rub the rag over the adhesive to remove as much of it as possible. Scrape off any remaining adhesive with a putty knife.
Peel the backing off of a new peel-and-stick plastic tile, or spread a thin layer of plastic tile adhesive on the back. Place the tile, sticky side down, over the spot where the scratched one was. Press down. Place a brick over the new tile and leave in place for 24 hours so that the adhesive can set.