If I were to sign over parental rights and my name is not on the birth certificate and I have not been responsbile for this child, am I still liable to pay child support?

Are you the father?

Then yes.

All the mother has to do is go to court require a DNA test and demand child support.

With that, you also just can' "give up your rights" you would have to get the mothers permission and then get the judge to agree, seldom happens.

But if you do give up your rights, you are actually claiming the child, and may have to prove it is your child by DNA to give up the rights,
But your rights are visiting, having a say in the child's life.
That is what you give up.

Your obligations can not be given up, unless there is a new partner for the other parent who will adopt.

And of course can we talk about moral responsibility in raising and having a part in your own child's life.