Interesting question. I can only tell you what I know and that may not mean much! I am not a lawyer by any means.
By definition termination of parental rights means that the parent whose rights are severed has the same relationship to the child as a stranger on the street. There is no obligation of support. In my non-expert opinion, the birth father could not be expected nor required to pay support because his legal rights were severed by the court.
Just for thought though. When I was working for CPS in Oklahoma, we had a problem with parents relinquishing their parental rights because they did not want to pay child support. Most courts have ruled that a parent cannot relinquish their parental rights just because they do not want to pay their support obligations. The court won't accept the relinquishment in other words. There is a difference between relinquishment of parental rights and termination though.
Also, if the parent's rights are terminated or the relinquishment is accepted by the court because of other circumstances not solely on the lack of support then the back support owed is not forgiven. The arrearage is still required and legally binding.
Also, from what I understand, the matters of custody, parental rights, termination, and such are functions of the state judicial system so there could very well be differences from state to state.
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