If the child is adopted (for example, by mother and her spouse), then you have no legal duty to support the child. Otherwise, the answer is no.
This is my general comment only. For legal advice upon which you can rely, you should contact me or another attorney and provide more information about your situation.
Most states will not let you relinquish parental rights unless there's an adoption pending. Courts view the situation as the child being the one entitled to support (not the other parent) and to knowing who his/her parent is. You and the mom can also forget getting a court to let you forego visitation in exchange for not paying child support. There's no trading visits for support (or lack thereof). Best you can hope for is that she meets someone great who wants to adopt the child.
Unfortunately, child support is a poorly publicized consequence of unprotected sex.