Termination of the rights of a unborn child

The only way you can to terminate your son's legal responsibilities is with a DNA test that subsequently proves the child was not fathered by your son. A court is not likely to order a DNA test until after the baby has been born. Instead of getting your son out of this situation early, you should be letting him experience the stress and uncertainty he brought upon himself by allegedly having unprotected sex with the girl. If he is exonerated by a DNA test, the stress will be alleviated. If the test shows he is the father, then he deserves everything he's been through plus additional sanctions for lying to you in the first place.

If the mother of the girl adopts the child, that should cut any ties your son has to the child.

however, if the girl keeps it, and the mom DOESN'T adopt the child. and raises it. then your son (and you until he is 18) are liable for child support payments

If he doesn't sign the birth certificate and she doesn't name him as the father then there is nothing you need to do. She will sign the surrenders or consent for adoption. If he signs the voluntary acknowlegement of paternity wich they would make him sign if he was at the hospital and wanted to sign the birth certificate, then he would have to wait 72 hours after a baby is born then he could sign a termination of parental rights or they call it a voluntary surrender or consent for adoption.
He would say that it is in the best intererests of the child. If he doesn't sign the birth certificate then she would have to prove that he is the father before she could persue any action.