Dark Hollow Tr, the Big Meadows Nature Tr and the climb up Big Rag in SNP also don't allow dogs although I've seen people take dogs on Old Rag.
If your dog hasn't been active keep in mind that they have to get conditioned to hiking just like humans. Avoid overly rocky trails. It will tear up their paws. I personally wouldn't take a dog anywhere that rock scrambling might be required. You will have a hard time getting an injured canine out.