Move the furniture to a work area. Ideally, because of the dust and fumes, this project should be completed outside. If that’s not possible, place drop cloths underneath the piece and make sure your work area is well ventilated.
Sand through the chew marks. For deep areas, use an electric hand sander on a low setting. For slight marks, regular sandpaper will be sufficient. Use a medium grade at first to get through the marks and then sand with a fine sandpaper to even out the finish.
Fill any deep holes with wood putty. If the dog's teeth left deep holes, patch these with wood putty. Pick a color close to the color of the wood and use a plastic putty knife to smooth the putty into the hole. Scrape away any excess putty immediately. Sand over the patched area with fine sandpaper when the putty has dried.
Stain the patched area. Use a stain that matches the rest of the wood. You may need to use more than one coat to match the color. Sand the area lightly when you are done.
Seal the wood. Use a clear polyurethane to protect the patched area. Apply one coat, let it dry, sand it lightly and apply a top coat.