D&D 3.0/3.5
It now does:
It takes time to put on armor.
Armor makes you slower. (There are now three "classes" of armor: Light, Medium, and Heavy. Medium armor reduces your Speed and Heavy armor also makes it impossible to run.)
Hm. By the way, I'm pretty sure that heavy armor reduced your speed even in some earlier editions of the game...
The heavier and bulkier the armor, the less you can depend on your reflexes for protection. (Every armor has a Maximum Dex statistic. This is the maximum bonus Dex can give you to AC when wearing that kind of armor - no matter how much your actual Dex score would give you when unarmored.)
Armor encumbers you in other ways. (E.g., penalties on certain skills. Heavier armor types generally give you higher penalties.)
Arcane spells (e.g., spells cast by a wizard) have a certain failure % chance depending on armor type.
An unarmored combatant avoids these problems; a lightly-armored one, most of them.
Well-made ("masterwork") armor has a slightly lower penalty to skills. And armor made of special materials (e.g., "mithral") often has even less disadvantages.
Anyway, some examples:
Chain shirt: AC Bonus +4; Maximum Dex Bonus +4; Armor Check Penalty -2; Arcane Spell Failure 20%; Speed 30' (assuming you're a human, elf, orc, etc.)
Chain mail: AC Bonus +5; Maximum Dex Bonus +2; Armor Check Penalty -5; Arcane Spell Failure 30%; Speed 20'
Full plate: AC Bonus +8; Maximum Dex Bonus +1; Armor Check Penalty -6; Arcane Spell Failure 35%; Speed 20' (and can't run)
So a character with an 18 Dex (+4 bonus to AC) might prefer the chain shirt; a character with a 13 Dex (+1 bonus) might want a heavier armor. Might.
There's more to D&D 3.0/3.5's AC system, of course - this is only a summary of the most important stuff.
Now, one more consideration of note for the "light fighters vs. tanks" debate: Armor doesn't help against "touch attacks" (i.e., attacks that only need to, well, touch you to be effective) and your Dex bonus doesn't help in certain circumstances (e.g., when you're surprised).
So... Yes, there are now some distinct downsides to being a tank in D&D. Still, heavy armor is hard to beat when it comes down to sheer protection value - the only question is if you're willing to pay the price.