It is possible that the property doesn't sell during that time. Sometimes, the owners are able to do a modification or just catch up on their payments and the house doen't make it to the sale. If the sale takes place, the new owner will have the option of evicting the occupants or make arrangements for a rental. It would all depend on what the new owner intends to do.
The chances that the mortgage company will let the tenants stay in the house are null. Banks are not in the property management business. They would rather have the house empty for years, let it get vandalized etc. than have a paying tenant stay in the house.