The loss of lives, hands down. While I believe there bad people in each of those places you mention, I also believe that there are innocents who are as much a victim of their evil as the people of Iraq were victims of Saddam. The people of New York, the Pentagon, and in a Pennsylvania field on September 11, 2001 were victims of this same brand of evil. And I would not have advocated shooting them out of the sky to save a museum either were there a chance a single one of them might have survived the crash. More to the point, I would not have advocated placing a platoon of soldiers in front of that same museum to stop a speeding bus from crashing the building.
The oil fields are being protected because they are the life blood of the Iraqi people. They will provide the revenue they will need to rebuild their country and crawl out from decades of poverty and repression. Having said all that, the article came from (The Sky is Falling) Reuters. There are other articles posted right here on FR indicating that Iraqi officials were in charge of that building and had been trained to remove those artifacts should war begin. I believe we will find that that is the case.