can a father sign over his rights to avoid paying child support?

Unless there is someone stepping in to adopt the child, the father may be able to terminate his rights but not his obligation for child support

Before an order terminating a parents rights may be granted, the judge must find such an order to be in the child's best interest. In my experience, most judges do not find a termination of parental rights to be in a child's best interest in such a circumstance unless there is another man standing willing to adopt and support the child.