What looks like a mere table to you may seem far more fascinating to your puppy or teething toddler, but the results of that fascination may not go well with your silver tea service. When the marks merely dent the finish, they aren't difficult to repair, but if the teeth marks sink into the wood, try to get them out by sanding or using wood filler.
Dents in the Finish
If you're lucky, you caught your puppy or toddler while they were just tasting, and all you have to deal with are some surface indentations. If the finish is lacquer, which is true for most interior furniture, you can make a simple repair with clear nail polish, which is also lacquer. Dab a little on the dent with the brush that comes in the bottle; wait for a few minutes; and, if it forms a depression; add a little more. When the dried surface is convex, you can flatten it with 600-grit wet/dry sandpaper lubricated with a little water, and you are ready to polish the wood.
Dents in the Wood
Teeth marks that affect the wood are a little more difficult to remove, depending on how deep they are. A procedure that often works is to swell the wood by covering the defect with a towel, setting a clothes iron on full steam and laying it on the towel. Two or three minutes of this treatment should raise the wood. If the bite mark was deep enough to discolor the wood, you may be able to find a felt marker with a color close enough to the stain for touch-up. Otherwise, dab on a tiny bit of stain with a cotton swab; touch up the finish by dabbing on nail polish and sanding with 600-grit wet/dry sandpaper.
If you were blissfully napping while your puppy was feasting on your favorite furniture, a few cosmetic fixes may not be enough to correct the damage. You can do a spot repair, without the need to strip and refinish the entire piece, by sanding the affected area with 120-grit sandpaper, a paper that is coarse enough to remove the finish and smooth the wood. A rotary tool with a sanding accessory helps if you're dealing with spindles and carved features, but use a low speed to avoid changing the shape of the wood. After smoothing with 150-grit paper and dabbing the stain with a rag, three coats of spray lacquer should complete the repair.
Calling in the Cavalry
You left your puppy home while you went to the movies, and when you got back, the corner of your table had all but disappeared. If you're dealing with painted furniture, you can call in the cavalry, and it will bring epoxy putty to your rescue. After mixing this rock-hard wood filler with its hardener according to the instructions on the label, you can mold it almost to the original contours of the damaged area. When it sets, a thorough sanding completes the molding process; when you repaint a properly completed repair, the damage is history. This repair also works on stained-and-finished furniture, but the patch remains visible.