If a parent gives up custody of their child,are they still liable to pay support?

First, please pay more attention to posting guidelines. On the top of the Divorce forum, where this was moved from) is a Read First Sticky which directs questions of a legal nature to this forum. Had you done so, you would have seen some other Read First stickies that might have answered your question.

Giving up custody does not release you from responsibility to support your children. But I think you meant giving up your rights, not just custody. The answer to that is that you can't give up your rights. Only a court can terminate parental rights and they will only do so in specific circumstances (as outlined in one of those stickies). They will NOT do so to relieve the parent of the responsibility to support their children.