First, continue paying rent even if your landlord dies. If and when the home is foreclosed upon your boyfriend is protected by the Federal Tenant Protection act of 2009. After the bank acquires title to the home he will have up to 90 days to vacate. Although the bank may offer cash for keys.
If the landlord dies, any lease remains in full force and effect until its original expiration debt. The tenant will pay the rent to the owner's estate which will manage the property, make the mortgage payments, etc. It's actually rare for a home to go into foreclosure because the landlord dies, assuming that the executor of the estate is on the ball.
If the property goes into foreclosure, any leases die with the foreclosure but any tenants have a minimum of 90 days to move once the new owner (the lender in most cases) gives them notice of the foreclosure. Many lenders will offer "cash for keys" to entice tenants to move early. That's optional if all parties agree and the agreement is in writing.
You mention a reverse mortgage. Does the landlord still live in the property? Normally a reverse mortgage requires that the owner occupy the home as his or her principal residence. If the owner moves out, the mortgage becomes due and payable in full, typically within six months.
At any rate, a reverse mortgage is not much different from a conventional one once it comes due. Either the property must be sold and the loan paid off from the proceeds, or the lender forecloses on the property. The impact to any tenant is the same. If the property is sold, the lease remains in full force until its original expiration date. If the property is foreclosed, the tenant must be given no less than 90 days to vacate.