You must continue to pay rent or the owner can (and probably will) file an eviction notice. I see it happen all the time on short sale rentals. Some unethical owners collect the rents and never pay the lender and just pocket the money. Other owners do make their mortgage payments so I don't want to say your landlord is one of the sleazy ones.
As long as you continue to make your mortgage payments you can stay. You will probably have to allow the place to be shown to prospective buyers. The short sale can get approved as soon as a few months but I believe you are able to stay until your lease expires--as long as you continue making your rental payments.
Kevin Flynn Realty, Inc.