I am not sure what you mean by statute of limitations on a criminal fine. If you had a suspended sentence, your case would be closed at the end of the jurisdiction period, with any remaining fines being sent to collections. If you want proof of payment, your best option is to contact the court to verify that the fine was paid in full and get proof from the court.
To answer the question posed in your caption, the answer is NO. A fine is imposed as part of a sentence. Once imposed, it remains a part of the sentence, expiring only when it is satisfied. Good luck.
This response does not constitute legal advice. Given the nature of this website, it does not create an attorney-client relationship. This answer is provided solely for informational purposes, for you to use as a starting point when speaking directly with a lawyer in your State. I urge you to immediately contact an experienced criminal defense lawyer admitted to practice law in your State before you make any decisions about this case.