If a father gives up his rights does he still have to pay child's support

Not if its done through adoption. If not then he still is liable for support.

Well you start with a false assumption. That a father can give up his rights.

I suggest you read this sticky note:

https://www.askmehelpdesk.com/family-law/signing-over-rights-read-first-116098.html

If you have any more specific questions feel free to ask.