If a man goes to get a paternity test on a child unknown and finds that it is his, can he sign his rights away and not be obligated to pay support?

A parent can always offer to surrender their parental rights to a child.  The process usually involves signing an affidavit of relinquishment and filing a petition to terminate.  However, the final decision is with the judge.  The judge will base his decision on the needs and best interest of the child.  If someone else is willing to step in and be dad, then courts tend to be more willing to grant a petition to terminate.  They are also more willing to grant termination if the parent has done something to harm the child. 

If your only reason for termination is that you don't want to pay child support and the mother is opposed to the termination, then there is a good chance that the judge will not grant the motion. (The policy reason is that they don't want a child to be in the world without a father of record unless they just have to)  If mom is good with the motion and there is a new guy in the picture, you'll have a better chance of successfully terminating your parental rights.  Either way, if you can get mom to agree to the termination, you'll significantly improve your chances of having the motion granted.