Vacuum the spot with a vacuum that has a beater brush. Brush the shag pile around the burn mark out, away from the mark, using your hands.
Cut around the burn mark with a carpenter's knife. Cut a square around the mark so you can easily cut the patch to fit.
Pull out the cut-out portion of the shag. Turn the patch of extra carpet over, wrong side facing you. Place the square of burnt carpet on the edge of the patch and trace the other sides with the carpenter's knife so you have an identically sized and shaped patch.
Run a bead of hot glue along the edge of the patch, 1/4 inch in from the edge. Run a "x" shape across the middle of the patch.
Turn the patch right side up and press it into the hole in the shag carpet. Let it dry for 24 hours, then vacuum again to raise the pile so it matches the surrounding shag.