No, It will half close and shudder a bit, then open, without ever completely closing.
I only remember this because I very recently played through MI2.
Also, what would be the point of the balloons if the door closed without them? There is nothing important in the Melee alley. If the doors could close completely without the balloons, then the balloons would just be a superfluous obstacle to getting nowhere.
God is a placebo with nasty side effects.