Vacuum over the burn mark with a hand vacuum, removing any loose mattress fabric as well as dust.
Fill a mixing bowl with 1 cup water and 1 tsp. liquid dish soap.
Blot, but do not rub, the burn liberally with the mixture, using a clean cloth. Rubbing may cause tearing or smearing.
Cut any unraveled fabric away from the mattress with a razor around the burn mark.
Tuck the burned fabric into the mattress, then pinch the remaining exposed fabric together. Hold it between your fingers and sew up the spot with a needle and matching thread to finish.