Armor class never has anything to do with the damage you take,1 only whether or not you get hit in the first place. The only thing AC is ever compared against is an attack roll, not damage taken. So in the strictest sense, we could say “no,” armor class doesn’t proect against damage—from falling objects or anything else. By the time damage is rolled, AC is irrelevant.
But to be less pedantic, we can look at whether or not AC helps you avoid being hit by falling objects, and thus avoid taking damage, reducing the expected damage you would take.
And the answer to that is...
...the rules aren’t very consistent.
The core rules on falling objects are frustratingly vague. They say
Just as characters take damage when they fall more than 10 feet, so too do they take damage when they are hit by falling objects.
Objects that fall upon characters deal damage based on their weight and the distance they have fallen.
[Several more lines and a table describing how much damage is dealt.]
Notably absent here is anything about how to decide if “falling objects” actually “hit” a character in the first place. Literally all of the rules focus purely on the damage dealt once you resolve a successful hit.
There are some precedents we can look to. Cave-ins allow a DC 15 Reflex save to halve or negate the damage (depending on where they were when it started). The “stone blocks from the ceiling” trap makes a melee attack with a +10 bonus to decide if a character is actually struck by a block. (A melee attack? That seems very odd to me, personally. Though I’m also not sure what difference it makes, since the +10 bonus is fixed either way.)
And then Heroes of Battle—decidedly not a core book—gives us more rules, specifically for “aerial bombardment.” That forces flying creatures who wish to drop heavy things on targets to make an attack roll against AC 5 to hit a particular square, and then whoever’s in that square has to make a DC 15 Reflex save or take damage (success means no damage).
So a DC 15 Reflex save to avoid something falling on you seems to be more-or-less the norm. Armor doesn’t help you at all with Reflex saves, so the answer to your question, in that case, would be no, armor class does not protect against damage from falling objects. But in the “stone blocks from the ceiling” case, where the trap is making an attack roll, armor class does protect against that, so there the answer would be yes.
- Unless you are using the armor as damage reduction variant, in which case armor actually does reduce how much damage you take. But not from falling objects, probably—DR only reduces damage from weapons and natural attacks, and falling objects aren’t either of those, except maybe they kind of are? Especially when they’re purposefully dropped on targets, they seem more like improvised weapons than not to me. Anyway, the rules never categorize falling objects that way, so as written, the rules for DR would have it do nothing against falling objects.