If my landlord is no longer paying his mortgage, do I need to still pay rent?

Yes, per your lease agreement, you are required to uphold your end of the bargain, that is -pay rent to the LL.   In the event the current LL is unable to rectify his issues with the lender and the lender forecloses and either sells the property to new buyer (new LL) or becomes the new buyer, you owe that new buyer the rent and unless the new buyer plans on living in the home, you have to the end of your lease term or 90 days, which ever is longer, to remain in the home.    

You are still required to pay rent to the landlord until they are no longer the legal owner.  This occurs when title passes to a buyer at the foreclosure sale.  The landlord's financial problems have no impact on your legal obligation to fulfill the terms of your lease.  If you do not pay your rent you can be evicted, taken to small claims for the money you owe, and have your credit report affected.  You should be aware that pursuant to federal law a lender who takes possession of a property or a new owner who buys the building at auction has to let a tenant stay for 90 days from the sale or the remainder of their old lease, whichever period is longer.  The rules are a bit different if someone is buying the property to live in; in that case, they can terminate a lease with 90 days notice.  Additionally, your landlord is still responsible for the return of your security deposit.  Under the circumstances, see if your landlord will let you apply it to a month's rent.  Again, stay on top of when the sale date is, after that you can stop paying your landlord (although you will then have to pay the new owner).