The right to collect unpaid child support does not go away. Illinois’ divorce law permits collection of past due child support at any time without limitation and owed child support does not even get discharged in bankruptcy.
If the paternity is in question, then there may not be court ordered child support due. Your question is very vague in that regard
One added note, even if the DNA test proves that he isn't the father, it may not be enough to undo any court ordered support. However, there's not enough information here to point you in any direction. Revise and repost with a more detailed history.