Allow the burn mark on the linoleum to cool, if still hot.
Rub a piece of 0000 steel wool over the burn mark. Move the steel wool with the direction of the grain. Continue to rub until the burn mark is worn off.
Remove dust from the linoleum surface with a hand vacuum.
Apply linoleum floor finish with a paintbrush, one coat at a time. Use the finish to fill in the space the steel wool created. Let each coat dry before adding the next. Continue to add additional layers until the spot is filled in. Allow the last coat to dry fully.
Sand down excess floor finish so the floor is level. Clean the floor with a damp cloth and one drop of dishwashing detergent.
Rinse the floor with a damp cloth. Dry the surface with another cloth.