Mom's in a nursing home and her residence is not paid for. They tell me I can't rent it nor let it be foreclosed. What are my options?

I'm assuming she's on Medicaid, so I'd call the Medicaid office. They will likely let you sell it and pay off the loan, then profits would go to your mom's care. Every state is a little different, so checking locally is best. You also could ask advice from an elder law attorney. They know all of the ins and outs of Medicaid.
Good luck,
Carol