You CANNOT sign over your parental rights unless another man is adopting your child as his own. The adoption will terminate all rights you have to even see your child. Any back support will still exist (and should be specifically waived via an entry drafted by the child support enforcement agency).
However, absent the case of an adoption there is NO way for you to "sign your rights over." If there was, every deadbeat father would be running for the nearest pen and the state would be supporting your offspring. Any contract you sign with the mother will not be enforceable. She can always hit you up for child support because child support is for the child, not the mother and therefore the mother doesn't have an absolute power to waive it. Furthermore, if she ends up on welfare, which most single mothers do, you'll be hit up for child support from the State of Ohio, and she won't have any power to stop it.
On the other hand, you singing a document saying you don't want to be a part of your child's life for financial reasons could certainly be used to show what kind of a father you are in a court of law. Just because you pay child support doesn't mean that you have a right to see your child... get the picture....
So absent an adoption situation with a proper waiver from the child support enforcement agency, should you sign such a document? Um.... no.
It won't effect your child support since there is no such thing.
You may be able to enter into an agreed order saying that there will be no child support and a no visitation order. These are dangerous as either of you can in the future go back and file to change this. Also if she would go onto public assistance, the State of Ohio will seek child support payable to the State to reimburse the public assistance payments.
It is best that you stay away from this situation and just continue to visit with your child as that is what is likely is best. There is also an argument for a downward deviation in your support if she is representing that she can provide 100% of support full time for the child without your assistance (provided she is not on government assistance).