After a father relinquishes his rights, is he still required to pay child support?

Simplily put YES

Yes if the father merely gives up his "rights" he just can not visit, he can not have a say in the raising of the child. But it does not take away any of his obligations to pay child support or furnish health insurance.

The only normal exception to this is if the parent with custody remarries or has a partner and they adopt the child. Then the support payments stop.

But think seriously, would there be 1000's of men sitting in jail this morning for being late or not paying support if they could simpliy just sign away that.