If your finance company refuses to cash the money for a debt owed, do you still owe them?

That is not the question. There are a whole lot of details missing here. What is this for? Why is the money owed? Why was cash refused? Was the amount of cash tendered the exact amount of the claim? Since you post no details, you either need to re-post your question with the relevant information or you need to talk to an attorney. There could be a possible defense if you would be sued but more information is needed to establish this. If tender was not proper then you would still owe the debt.