Hi! True linoleum takes paint quite well. Most flooring today that we call linoleum is actually vinyl. You can tell the difference between the two because vinyl is shiny where linoleum has a matte finish.

In order to paint a linoleum floor you need to remove the old floor wax completely for the paint to adhere properly. You can find wax strippers at most hardware stores. Once the wax is removed, lightly sand the area with a medium grit sandpaper and vacuum thoroughly prior to painting. Paint with oil based paint and be sure to use an oil based primer. After the paint has COMPLETELY dried, very lightly sand the paint and seal it with polyurethane. Polyurethane comes in different finishes from matte to shiny.

To paint vinyl, first clean the area you want to paint with TSP (tri-sodium phosphate). Sand the floor in order to remove the glossy finish. If you have tiny ridges in your floor you probably won't be able to remove all of the finish. You can purchase a product called a Deglosser from a paint store to completely remove the finish. Lightly clean the area with a damp cloth and let dry. Paint and prime with oil based paint and seal with polyurethane.

Another idea would be to patch the floor in the area where it was burned. Cut out the burned section and patch it with a matching section taken from a corner or some other inconspicuous area of the room.