The Obamas paid $45,046 in mortgage interest in 2012, which appears from the disclosure statement to be at a 5.625% interest rate with Northern Trust. That suggests an outstanding principal balance of about $800,000.
On the other hand, the bulk of their investments are in Treasury notes. Based on the disclosures, I estimate they hold about $3 million in Treasury notes (also held by Northern Trust), yielding 0.71% if averaging a five-year maturity.
By selling some of those Treasuries and paying off the mortgage, they would effectively be getting five more percentage points on the amount; they would also be about $40,000 better off each year before taxes, not to mention being less exposed to notes that could take a hit from possible rising rates.
The Obamas would pay more in taxes but make much more after taxes -- especially since they aren’t getting the full deduction anyway, due to the AMT. That's more money going to the U.S. Treasury and more money for them; Northern Trust would be the loser.