Finish? You might try stainless steel. Sorry, but a wooden door is quite a bit softer than your dog's nails, and if that door is like many, it is made of a relatively soft wood anyway. There is no paint, varnish, etc., that will stand up to a determined pet, because the wood underneath the surface coating will dent. Once the wood underneath dents/deforms, the surface coating is toast.
Better is to teach the pet not to scratch at doors. No, it is not always an easy thing to do. Like a small child, a bored dog looking for attention will even do things it knows to be wrong, as any attention is better than none. My approach was to give our previous dog a spot to curl up under my desk. He would happily fall asleep as long as he was touching my feet. Then later when I did have time, it was walk time or play time, and he knew it.