He will still pay child support unless the child is being adopted.
Paying child support will not cause the father to lose his parental rights - neither will not paying child support.
Usually this happens preparatory to an adoption. Signing over one's rights does not, in itself, terminate child support.
He can give up his parental rights (this usually happens as part of the adoption process), but his support obligation continues until/unless the child is adopted. There is no "right not to pay child support."
Child support remains.
An unmarried father cannot "choose" to not pay child support. The laws in every state require that a father pay for the support of his child. The mother must bring an action to the appropriate court so that a child support order can be established. Visitation rights are separate and a father can have visitation rights established by the court. Visitation rights are not dependent on paying child support.
if the mother terminates her rights can he collect child support from the mother if child lives with him?
How does he have any visitation rights with a custody and child support order?
Terminating parental rights does not mean you are no longer obligated to support the child so you still have to pay child support. Unless the child is adopted. You will have no rights to the child at all if you have your rights terminated and the court will not give them back so be sure what you are doing.
Signing rights away will not stop any child support from being taken from the father. If the child is the fathers then he is financially responsible for the child weather he has rights or not.