Do tape worms make dogs scratch and bite at their tails?

They can - passing the worms can be irritating, which most dogs would respond to by trying to scratch or bite their tails.

More likely, your dog may have another type of parasite such as fleas (which tend to accumulate around the tail head) or lice. If the irritation persists, or your dog's tail starts bleeding or scabbing over, I would suggest taking him to the vet for an examination.