Sand in and immediately around the damaged area with fine sandpaper -- 240-grit sandpaper is a good choice. Fold the paper and use the corner of it to get inside the gouge or burn, down to the surface. Buff it down to bare wood. Blow off the dust.
Apply wood filler to the gouge or burn with your fingers, packing it in tightly. Use the edge of an old credit card to scrape off the excess so it's flat.
Allow the filler to dry overnight. Buff it lightly with the sandpaper to get it smooth and flat.
Brush wood stain onto the filler. Let it sit for a minute, and wipe it off with paper towels. If the stain isn't quite dark enough to match the surrounding stain, repeat until it is.
Brush polyurethane over the repair area, matching the surrounding gloss. Let it set for a day.