Can a father in Tennessee release all his rights to a child because he doesn't want to have anything to do with the mother or the child?

An unmarried father in TN doesn't have any rights to a child unless he goes to court and asks for them. If he doesn't do that, no rights. His obligations, primarily child support, are separate and a person can't sign away their own obligations. He can see the child or not, but he will need to pay support.