I know this man that has a child. He lives in wa. And the child lives in Tn. He has never seen the child. The mother moved while she was pregnant. Gave the bany up to foster care, now the grandparents have the child and are seeking support from him. Can he sign his parental rights over so he is no longer responsible for child support

NO! There is this myth that a parent can just sign away their rights. But it is only a myth. He chose to have sex with this woman, now he has to deal with the consequences of that choice.

He should require a paternity test to make sure he is the father, but if he is, then he will be responsible for supporting HIS child. Whether he decides to be a part of HIS child's life will be up to him. If he wants, he can possible get full custody and definitely some level of visitation.

No he can not, he can file for custody of the child, since he is the father and ask for the child to be returned to him.

It is possible that he may not owe support. It mostly depends on the true process that has taken place when the baby was given up. Was the baby given up shortly after birth? Are the grandparents seeking child support from the mother also?