If my child is 15 and I've not seen him or supported him in 14 years, can I give up my parental rights?

Generally, a parent cannot give up their parental rights to their own child unless an adoption, whether this is by a third party or by a step-parent, has or will take place. There must be a willing and able parent ready to assume the responsibility. There must be good cause for a court to terminate the rights of a parent. Not seeing or supporting a child is not a good cause reason to terminate parental rights. Communication of information by, in, to or through this Web site and your receipt or use of it (1) is not provided in the course of and does not create or constitute an attorney-client relationship, (2) is not intended as a solicitation, (3) is not intended to convey or constitute legal advice, and (4) is not a substitute for obtaining legal advice from a qualified attorney. You should not act upon any such information without first seeking qualified professional counsel on your specific matter.

It's possible. But you can't.

You can always agree to give your custody and visitation rights to the other parent. For example you can agree that you are to have no visitation and sole legal and sole physical custody to the other parent. There is no process for you to terminate the fact that you are the parent unless there is another person (usually a step-parent) who adopts the child.

Not unless someone is going to adopt him.

Some courts are reluctant to allow this but after that long period of time it would certainly be worth trying.