Here's for starters - I'm surprised you didn't know. Did you read the instructions before you signed?
How to Correct a Birth Certificate in Ohio | eHow
Has the man on the birth certificate paid support, expenses, anything else? He has legal grounds to pursue you for reimbursement. These "you're not really the father" cases do not always go well for the mother.
So pursue the DNA, determine the ID of the father, change the birth certificate, pursue the father for support.
If the State/County paid ANY of the expenses in connection with the birth the State/County can look at the father for reimbursement. If they have targeted the wrong man, there will be problems.
I had one of these cases in my extended family in NY - child was 2, father did not live with the mother but he AND his family were in the child's life, and then the mother "decided" that someone else should be tested, and my family members was not the father. Why/how someone could lie like that, I have no idea. She also didn't want the child not to have a father so the family member was put on the birth certificate - being a father has nothing to do with the name on the because, by the way. There are great stepfathers, adoptive fathers, other fathers out there who are not the birth father.