US has might, influence and power. Qatar has strategic location and a crapload of natural resources. We both help each other for selfish reasons.

US isn't fond of Iran, for a wide variety of reasons (Islamic revolution when they held hundreds of our citizens hostage for over a year; their calling for the destruction of Israel, etc.).
I don't think Qatar is crazy in love with them (if you think the Sunni in the Gulf region aren't collectively scared sh*tless by the spreading influence and power of the Shia, you're smoking crack..), but they realize the benefit in making nice with them and the important role they play in the region.

The Israelis were terribly worried when Obama won the US election, that he would warm up to Iran a lot more than McCain ever would consider doing. It's an honest concern, because I think Obama WILL engage with them more than McCain ever would have.
But what the end result will be is anyone's guess...

Qatar and USA will work out a mutually agreeable solution if there is a rift between them, I have no doubt.

"Most plain girls are virtuous because of the scarcity of opportunity to be otherwise."
-- Maya Angelou