I think - war or no - we have an obligation to do everything we can to preserve our collective heritage. These sites are quite simply irreplaceable. One of the things that traditionally separates barbarians from civilized peoples is the ability to recognize, even during war, that there are some things worth saving. To allow a place like Babylon, which is one of the most important archaeological sites in the world, to be destroyed is unconscionable. To contribute to that destruction is to become the barbarian.
There are traditional roles even in warfare, and particularly in western civilizations' perspective on war - many of them borrowed from the medieval chivalric codes. The good guys wore the white armor (or the white hat), fed and protected the innocent and noncombatants, offered medical aid to all who needed it, and did everything they could to salvage the heritage and culture around them. There were rules about the humane treatment of prisoners and many other similar issues. Observation of these rules made it harder to fight against the bad guys - because they didn't observe the rules - but the mere fact that they didn't observe the rules is what made them the bad guys.
If we begin to ignore these rules and sacrifice everything by which we have been governed for so many centuries - what have we gained? If we stop observing the rules in order to win - who are the bad guys? Are we winning anything or is it time to question who we have become?
I find myself bemused by the world's ability to spend millions on museums, research and preservation, and yet stand by and allow something like this to pass with barely a word. Then again - who would listen? Barbarians get their name from their inability to communicate intelligently.