The hand-over of sovereignty seems to have been highly cathartic in Iraq. The US reputation has soared in that country for the moment but there is still an opportunity to screw things up royally. At some point, there is going to be a serious disagreement between the US forces and the Iraqi government. Perhaps rebels will attack US forces and the Iraqis will demand that the US not pursue them. Perhaps the Iraqis will demand that the US forces vacate a position that is necessary for security. There are many opportunities for conflict between a local government and a large foreign military force. How the US responds to these conflicts is going to be pivotal.
If the US ignores the express demands of the new Iraqi government then this will constitute evidence that the US really is an invader. Fair or not, that's how it will be perceived (and it's more fair than not). If the Iraqis really are sovereign in their own country, then the US forces are present at their sufferance. The US should not forget that. If the Iraqi demand is anything that the US can live with, then they should live with it. The US owes it to the host country to respect their rules.
If it's something that the US cannot live with, then the US has an important bit of leverage, an ultimatum: they do what they need to do or they pull out completely. This is a last resort because once they say it they have to stick by it. The ultimatum will likely motivate the Iraqis to find a way around the impasse. If not, then the US has to pull out. This could lead to civil war in Iraq, but it's their decision. And we can hope. Maybe there would be no civil war. Maybe Iraq would stand up on its own as a fine new democracy.
This could even end up being a great good. If the US is forced out of Iraq in this way (or even by a direct demand of the Iraqi government) then the world will see a Muslim proto-democratic government succeed where the most feared Muslim dictator and the most feared Muslim terrorist failed. And failed miserably. Democracy will be the new strong horse. Ultimately, that could be the best of all possible outcomes.