I want to round up some of the main arguments that have been employed on this question. It would be fair to say I feel ambivalent about it, but I'm still chewing it through (which is, of course, what a blog is for....)
Argument #1: it will endanger the troops (aid and succour the enemy)
On one level this is facile, in that, in a war, we assume that the enemy is trying to kill our soldiers already. It is not facile in that burning the koran will reinforce the ideology and remotivate their troops. So the good point here is not about fear but about pragmatism - you don't give the enemy a propaganda victory.
Argument #2: it will endanger other Christians around the world
This may be true, but if so, it actually says worse things about Islamic culture, ie that a symbolic protest such as this might lead to loss of life, because the values and cultural norms are so uncivilised in such countries. So to not burn the Koran for this reason alone is simply to succumb to intimidation.
Argument #3: it's rude and disrespectful
It undoubtedly is rude and disrespectful - it wouldn't be worth doing if it wasn't - but I'm not sure that 'rude and disrespectful' automatically make something wrong, it depends upon everything else.
Argument #4: it's symbolic violence
Yes it is; it is not a peaceable act, it is not something that will generate good will and foster further understanding. However, again, I'm not sure that there isn't (in principle) a place for symbolically violent behaviour. Whether such behaviour is defensible or not depends entirely on the wider context - is it simply bullying or spiritual abuse? Or is there a wider toppling of idolatries going on (leading to less abuse)?
Argument #5: We are told to love our enemies and this isn't loving (Jesus wouldn't do it)
If we take the feelings of the targeted audience as the end point of the process then "causing pain = not loving" follows. Jesus, however, often had a further end in view relating to the long term liberation of the people he engaged with. What this action does is totalise the argument. It's an action which follows once reasonable discourse has come to an end (or not been started). The real question is one that Byron asked, about whether a creative and attractive alternative is being shown. The action is defensible to the extent that creative possibilities are held out (which is something that doesn't seem to be the case here).
Argument #6: it's ugly and stupid and childish ('Ugh!')
This is an expression of our cultural norms, and whilst I tend to the view that the argument from disgust shouldn't be rejected on principle, I don't see what it adds here. In principle (I'm abstracting again) it could be the cultural norms which are the idols needing to be toppled, just as much as any Islamist nonsense.
Argument #7: how would you like it?
What's sauce for the goose is sauce for the gander. The difference is that the power of the act would be different, given that Christians don't see the New Testament (fundamentalists excepted) in the way that the Islamists see the Koran. Of course, what we do see as the Word of God was treated in just such a way - he was crucified - and therein lies some of the most important differences...
Argument #8: it's retaliation
As a motive, I would agree that this is wrong. I'd simply point out that burning books is not on a par with burning down buildings by flying planes into them.
That's my thinking so far; I haven't finished yet.