If a father gives up all rights to a child does he still have to pay child support or back child support?

Whatever makes you think you can give up your rights? This is a myth. Only a court can terminate parental rights and they are reluctant to do so. They will NEVER issue a TPR just to let a parent out of paying support.

Some states do terminate only rights and not responsibilities, but even in those areas, TPRs are hard to get.