This may sound like a stupid question, but here goes:
What happens if the financial institution to which you owe money goes under?
Will the money still have to be paid to someone?

Technically, yes. There are lending institutions that go bust; 'Goldome' comes to mind; in the 1980's, Goldome made a lot of home improvement loans. The accounts receivable may be found, or they may not. The debt is owed, never the less; the wise course would be to save your payments in an interest bearing account until the opportunity comes to pay it off.