when you relinquish parental rights you are giving up all claims to the child, emotional and financial. you will have no voice in how the child is being raised, you will have no rights to see the child, and you will have no financial repsonsibility to the child. (no child support)
usually - unless there are special circumstances - this can only be done if someone is willing to adopt the child and assume the role and responsiblity of being the father.