Lay a linoleum patch over the burn area. Use a patch that is larger than the burn area by at least a couple of inches on all sides.
Line up the patch so the pattern matches exactly with the surrounding flooring. Tape the patch down with a strong tape once the patterns match.
Position a straightedge, such as a ruler, over the patch. Cut a square or circle through both pieces of linoleum. Make the area large enough to cover the burn spot plus about an inch or more on all sides. Use caution to cut through the linoleum vertically. If you slant the utility knife this may cause a gap in the patch.
Lift both pieces of linoleum. Discard the burned piece.
Set the linoleum patch down into the opening to ensure it matches perfectly. If not, cut another patch.
Cover the floor where you require the patch with linoleum adhesive. If it is a large area, use a V-shaped trowel to apply the adhesive; otherwise, squeeze out a stream from the package. Add the adhesive in the center and near the perimeter of the floor opening.
Set the patch down into the opening and press it with your hands. Smooth over the patch to squeeze out any excess adhesive. Wipe up any excess with a damp cleaning cloth.
Roll a rolling pin or a hand roller over the area. This smooths out any air bubbles.
Set a heavy book or other heavy flat object over the patch until the adhesive dries well, according to the label directions.