A couple has a child out of wedlock. The father pays child support for several years, but the mother and child live in a different state from the father/where the birth occurred. The father wishes to terminate his parental rights in order to no longer pay child support. Which state's laws govern the termination of rights? Is it the one the child is a resident of, or where the child was born?
Generally, in non-emergency situations, the state with jurisdiction over parenting issues related to children including relinquishment and adoption is vested in the courts of the "home state" of the child.
A child's "home state" is defined by statute by a coordinated definitions set forth in parallel laws at the state (the Uniform Child Custody Jurisdiction And Enforcement Act in 49 states) and federal (the Parental Kidnapping Prevent Act) level. This definition of "home state" provides a fairly precise definition of something that approximately matches your intuition regarding the state where the child resides in most cases. There is also an international treaty with a similar home state definition.
The state where the child was born ceases to be relevant at the point at which that state ceases to be the child's "home state". But, once a state's courts take jurisdiction over parenting issues for a child, the barrier to divest that state of jurisdiction is higher than it would be if there was no prior litigation. So, the state establishing the child support decree would often be the "home state" in the fact pattern set forth in this question.
I will refrain from discussing the substantive law of voluntary relinquishment following an acknowledgment of paternity and payment of child support at length in this post as it is beyond the scope of what was asked. Suffice it to say that this is generally disfavored and is sometimes impossible. Also, the actions that could lead to involuntary termination of parental rights (even for mere non-support) also often have parallel felony criminal sanctions associated with them.