The landlord/owner owns the property until the foreclosure is final. Therefore, the tenant can live at the property at least until the owner loses possession. Also, if the owner has not assigned the rents to the lender, the he has no obligation to turn the rent over. If he did, it would only reduce the amount due the lender, and help to reduce any deficiency judgment (more owed on the property than the foreclosure sale brings - owner is responsible for the deficiency).
I don't know what state you're in, but most states have a residential landlord-tenant act which requires 30 days written notice to vacate the premises. Once the bank takes over, it would have to give this notice to the tenant if it wants the tenant out.
Please feel free to ask any follow-ups. Thanks.
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