If I am renting a house and the landlord stops paying their mortgage, do I still have to pay my rent?

Until title to the property passes your landlord is still the legal owner.  As such your obligation to pay rent continues.  Her hardship has nothing to do with you.

However, under the circumstances you may want to ask how she is handling the security deposit.  The new owner will purchase subject to you tenancy so you can stay until theend of your lease.  They will also supposed take over any security deposit account, so make sure that your landlord is planning on doing this.