Use 000-grade steel wool and a light touch to remove shallow burns. Sand only the surface of the linoleum.
Clean off all residue and sawdust with a clean, damp cloth.
Finish by applying a floor sealer to the affected area. Let the linoleum dry completely.
Cut a diamond-shaped repair piece and place it over the damaged area. Be sure to match the direction and pattern of the existing tiles. Secure the repair piece over the damaged area with double-sided tape.
Place a carpenter's square along one side of the repair piece and hold it in place. Score the edges not butted up against the square. Remove the repair piece and score the last side along the square. This method makes for a tight fit for the repair piece.
Cut the score lines completely through the damaged flooring. Cut down the middle of the damaged area and peel away the damaged tile, paying close attention at the edges.
Clean the old adhesive and residue away from the subfloor, using a wet cloth. Apply adhesive to the back of the repair piece and bend it into the opening.
Roll the repair piece with either a roller or a rolling pin to ensure proper adhesion with the floor. Clean any residue from the top of the repair piece, cover with waxed paper and weigh it down for 24 hours. Apply a floor sealer to the new area.