In my B/X house rules, any class can wear any kind of armor.
There are, of course, trade-offs though.
Encumbrance is the primary penalty from wearing armor, and this applies to all classes. I use the LotFP encumbrance rules, modified slightly to allow characters with exceptional strength to carry more. Fighters are, practically speaking, likely to have higher strength than other classes, so this also in effect makes them less likely to be penalized for wearing armor (though clearly this will vary on an individual basis). To maximize mobility, a prudent fighter will not carry much other than armor and primary weapons. Squires or porters can be hired to carry extra weapons and other equipment.
Wearing any kind of armor interferes with the delicate gestures required for a magic-user's spell casting. The following chance of spell failure applies: leather 1 in 6, chain 2 in 6, plate 3 in 6. If a prepared spell fails, it is not lost, but the round is wasted. The same goes for casting from a scroll; the scroll is not consumed.
Armor heavier than leather also interferes with the following thief skills: move silently, hide in shadows, climb walls, and legerdemain. Wearing heavier armor introduces a chance of failure: chain 1 in 6, plate 2 in 6.
Elves (or their human class variants, fighting magic-users) are not subject to spell failure when wearing armor. Faerie creatures such as elves are, however, not able to use mundane metal armor (such as armor of iron or steel) because close proximity of metal pains them greatly. They may use specially forged faerie-metal armor, which can only be procured in faerie realms. Elf armor tends to degrade if it spends too much time outside of elf lands, however. Dwarves are known for crafting durable faerie armor, but their prices tend to be steep and not just in terms of money. Dwarf-made armor is thus highly prized by elves that must journey in the sunlit realms.