Can a dealer make you resign loan papers if they say they made a mistake?

You should consult a local attorney, because laws vary from state to state.

In California, for instance, a dealer has a certain amount of time (a couple of days) to notify a buyer if a loan does not go through for some reason and instruct the buyer to either return the car or sign new paperwork for a different loan. If the dealer does not notify the buyer within that time period, then according to law the dealer must honor the terms of the loan as originally stated, self-financing if necessary.

However, in practice, the dealer may decide to reposess the car anyway and hope you don't sue, so unless you got a really good deal on the car originally, it's probably not worth the hassle.