Not directly, according to Bradford Plumer. But if oil prices trigger a recession, that will damage Obama:
[T]here’s ample reason to worry that surging oil prices could whack the U.S. economy. Higher gas prices at the pump tend to mean that consumers are sending more of their dollars overseas, leaving them with less money to spend on goods and services here at home. The Energy Information Administration estimates that a $20 increase in the price of a barrel of oil causes U.S. GDP to decline 0.4 percentage points.
So the real question is whether gas prices will crimp the broader recovery.
(Chart from Gas Buddy)