From the limited information that you provided, it sounds like this is a civil matter only. Unless the rental company has evidence that you stole the item, they cannot threaten prosecution to resolve a criminal matter. This would be in violation of the Fair Debt Collection Practices Act.
I would recommend you record the threat of felony charges and preserve the evidence to ensure you are protected. In the State of Ohio, it is legal to record a telephone conversation if one party consents to its recording.
If the Ohio rental company files a police report, then the State will determine whether or not to prosecute the case as a criminal matter.
Remember, you have the right to a lawyer and if the police come to discuss the matter with you, you'd be well advised to exercise that right. That said, it’s more likely that they'd file a civil action against you rather than file a police report.
Although I do not practice in Ohio, this threat from the rent a center sounds a lot like extortion. You may or may not remain liable for property stolen from you. If you have insurance, send the claim to the carrier. Also, call a criminal attorney and discuss the message with him or her.
The above is not intended as legal advice. The response does not constitute the creation of an attorney client relationship as this forum does not provide for a confidential communication.