If a father signed over his rights does he still have to pay child support if he is not yet on child support?

When a person has no rights to a child ("signed over his rights"), he also has no obligation. In other words, he would not have a child support obligation.

Under the law in Arizona the only action that will relieve a biological parent from the obligation to pay support is the adoption of the child by another. The adoption may be accomplished either by the consent of the biological parent, or by the termination of the biological parent's rights. Until the order of adoption of the child an action may be brought to require the payment of child support by the biological parent.