The fighter in plate armor is probably one of the most enduring tropes in D&D, and fantasy in general. The rules of the game are sometimes derided for a perceived bias toward heavily armored fighters, and all manner of "fixes" have been tacked on to rectify things. Sub-classes with special perks for going unarmored or lightly armored and arbitrary AC bonuses to make swashbucklers and barbarians the equal of their ironclad brethren are a common addition to the rules. There are probably, quite literally, dozens of iterations of those classes and others like them, both official and homebrew, floating around out there. Other times, gamers resort to cheesy Monty Haul assortments of magic items in their attempts at fighter parity - rings, bracers, and robes of protection, +5 leather armor, and so on.
In other words, they want to have their cake and eat it too. I think that's a mistake, based upon a mistaken perception. With apologies to Errol Flynn and Conan aficionados, from a purely combat-oriented perspective, the fighter in plate is, and ought to be, superior. In a straight-up, toe-to-toe melee combat, the lightly armored swashbuckler and the barbarian in his furry speedo are not the equal of the armored knight. All the energy used by the lightly armored fighter in his flurry of parrying and nimble evasion may indeed look cool, but that's energy and attention that the armored knight need not expend just to keep from being skewered. His passive defense, in the form of armor, allows him to focus more of his active attention on hitting and dealing damage. Between combatants of similar skill and experience, the armored knight is always the odds-on favorite. That's as it should be.
Well then, what's the point of playing a swashbuckler or a barbarian? fans of those archetypes might ask. They have a historical and a literary tradition behind them, and they're interesting from a role-playing perspective, but the rules give them the shaft!
In point of fact, though, it isn't the rules that give them the shaft. It's the focus of a particular game or campaign. In a game that heavily emphasizes combat, the traditional melee tank owns the field. Again, that's as it should be - if combat is your emphasis. There are plenty of situations, though, when heavy armor is either not useful, or a hindrance, and playing up those scenarios a little makes the lightly armored fighter or cleric a viable character.
One of the primary factors in the rise of swashbuckler and duelist-type fighters was not the effectiveness of fighting unarmored, but social reasons. Besides being uncomfortable and noisy, it might be a serious faux pas to go stalking around town in plate or mail. The overall effect would be not unlike that of making a trip to the shopping mall with a shotgun over your shoulder - it looks like you're looking to start trouble. It frightens the peasantry and arouses the suspicion (at least) of the guard and the local ruler. Going unarmored or discreetly armored and wearing a dagger, short sword, or rapier or carrying a quarterstaff is a better choice for the adventurer who wants to be dashing and prepared rather than seen as a violent brute or a paranoid cad. For adventures in urban settings, or the king's palace, a character wearing plate is going to be at a decided disadvantage.
Armor is great when you get into a fight, but for crawling about the dark depths of a dungeon or the wide wilderness, it can be quite a drag.
Time: Reduced movement rates are the most obvious handicap of heavily armored characters. Consider a lightly encumbered party with a movement rate of 120' per turn, vs. one with one or more characters in plate moving at 60' per turn. That's twice as long to explore 120' of dungeon corridor, and twice as many rolls for wandering monsters.
Fleeing or pursuing: If you want to run down a fleeing opponent, it helps if you're not carrying thirty pounds of metal wrapped around your body. And when you're fleeing from a combat that's too tough, well...the magic-user doesn't have to outrun the owl bear. He just has to outrun the fighter in plate.
Stealth and surprise: It's hard to sneak up on someone if you're clanking and jingling with every step. The standard chances for surprise - 2 in 6 - assume a typical party with typical equipment. If everyone goes in light armor, that could be bumped up to 3 in 6. Perhaps the chances of wandering monsters might be reduced as well.
Climbing, swimming, etc.: It should be nigh impossible to swim in metal armor, and it's not very good for scrambling over heaps of rubble, up or down cliffs and embankments, and into or out of pits and sinkholes either. Bogs and swamps are hazardous places, but even more so in armor. Full plate or mail in the desert or jungle is practically begging for heat stroke. That isn't to say that things can't be done in armor, but they may be more hazardous, make more noise, take longer, etc. Some places may be inaccessible, or nearly so, to heavily encumbered characters. (This obviously works best in a sandbox-style game, where there are no "plot essential" events or locations, only places that the PCs will have to come back to properly equipped if they wish to explore them.)
Carrying loot: It almost goes without saying that if you're lugging armor, it subtracts from your ability to lug gold and gems.
Skirmish combat: As I've already remarked, in a toe-to-toe fight, an armored fighter has a clear advantage, but there are tactical reasons to favor mobility over the protection of armor. Missile combatants, for instance, might find it more advantageous to use their unencumbered movement rate to stay out of melee altogether, and trust to cover for protection. A superior movement rate may be an advantage in cornering or surrounding a group of enemies. A fighter with a 40' per round movement rate can reach a beleaguered comrade 40' away in half the time it takes one with a 20' per round move to get there.
Once again, this is not to force any character not to wear plate mail, but merely to point out that there is, or at least should be, a downside as well as an upside to it. Does this diminish the special class advantages of the fighter and cleric? I say no. The thief and the magic-user are forced to use the light or no armor strategy. The fighter and the cleric can choose freely between heavy armor and light armor as the adventure demands. They can equip for stealth, exploration, and society, but the mage and thief can't deck themselves out for a melee-intensive grind. The ability to wear armor or not to is part of the fighter's versatility.
Tricking out swashbucklers and barbarians with AC bonuses and special abilities doesn't make them even with a traditional fighter; in a balanced campaign in which exploration and social adventures are on an even footing with combat, it gives them the unfair advantage.