If I purchased an item from a rent-to-own company and never signed a contract to make payments, do I still owe them money?

You are only obligated to pay what you agreed to pay when you purchased the item.  Please read the sales document carefully as it likely has payment terms contained in it.   If you paid less than full price when you bought the item, there was likely some payment plan to pay the balance.  However, if there are no such terms in the sales document, then you are not bound to make any payments.  You should contact the seller and ask what the payments are for.

Verbal contracts are enforceable, as are "implied contracts"--or contracts which are created from several different sources, such as advertising materials, bills of sale, terms on receipts and invoices, etc. Presumably, when you purchased these tires from the rent-to-own, there was some indication of the total price or the number and amount of payments to be made. You are bound to pay the advertised or otherwise indicated price for the tires even w/out a formal written contract. You do not have to pay more than would have been indicated as the price.