To cut to the chase, it really depends on what kind of lease you have and your local laws. Regardless of what type of mortgage the landlord has, whoever assumes the ownership of the home has to abide by the lease following the local laws (e.g. rent control). And it is never immediate; at a minimum, there would be a 30 day notice. If your boyfriend is on a one-year lease, the new owner (e.g. bank, son, whoever), must abide by the lease unless they plan on moving into the home themselves. If your boyfriend is on a month-to-month, then again following local laws, he may have 30-90 days.
Having said that, if your boyfriend is renting the house and the owner has a reverse mortgage loan on it, if the lender finds out that the owner does not live there, the loan will become due immediately and the lender may begin to foreclose. One of the very few conditions of a reverse mortgage is that the owner must live in the home as their primary residence. So they may not wait until he dies in order to collect on the loan (and/or foreclose) if he has not followed his contract.
I specialize in reverse mortgages (CA)