In NC, can a father sign over the rights of a child and stop child support? This father is not listed on the birth certificate and has not had DNA testing to establish that he is the father.

No.

And if there has not been DNA testing, how has child support been ordered?

Please read the sticky at the top of this forum about signing over rights and then post any additional questions.

Steve asks a very valid question. How was child support obtained if there was no establishment of paternity?

Not signing the birth certificate is no protection. But before a court will grant a support order, paternity needs to be established. Once paternity is established a court is not going to grant a TPR, especially not to allow the parent out of paying support.

By the way, there is a sticky note at the top of this forum that answers many questions about terminating rights. I suggest you read it.