Hi,
I may need to foreclose on my rental property that I have owned for 15 years.

The property appreciated over the years quite a bit. Since I am unable to sell the property in this down market and do a 1031 Exchange to shelter cap gains taxes, what happens if I foreclose? Am I still obligated to pay the taxes?

First, you don't foreclose. The lender holding the mortgage on the property forecloses. If the property has appreciated why are you having a problem selling?

As long as you are the owner you are responsible for the taxes.