Can father write off custody rights and still have to pay child support?

The state benefits you and your child are on matter a great deal. They are probably going to be a roadblock in the court ordering his rights be terminated, even if you are both agreeing to it, and even if he has no relationship with the child. The reason for this is because (to answer the subject question) terminating rights means he will not have to pay future support, and the court is unlikely to cut off an obligation to support when the state's providing you support. Now if you are married and your spouse is willing to adopt the child after the biological father's rights are terminated, then there may be some promise of this occurring.