If the father gives up his rights to his child does he still have to pay child support?

The laws differ from jurisdiction to jurisdiction. Some areas terminate both rights and responsibilities, some only terminate rights.

But in ALL places, getting a TPR is very difficult. Generally they are only granted to clear the way for adoption. A TPR is never granted just to let a parent get out of support.

So the likelihood is the father will not be allowed to terminate his rights. Or if he is allowed to, he will still have to pay support.