You can base the armor skill off of weight, and if the armor is of insufficient weight, it doesn't train armor user. Hence, light bone armor or leather that isn't sufficiently thick enough to really impair a dwarf won't train the skill.
Beyond that, if we get to the point where donning and doffing clothing could be routine and trustworthy, having military dwarves take off their armor to eat and sleep or when they are off-duty makes perfect sense, as well, since we, as players, aren't going to be bored or have to overly-micromanage that. (Although it will lead to the expected problems of an off-duty hammerlord with her hammer and platemail on the other end of the fort.)
Further beyond that, a clumsiness concept is worthwhile as an idea, and it's probable that Toady will, at some point, include something that simulates being clumsy in, but for now, simply moving and doing everything significantly slower when they are encumbered is a penalty worth noting. Weight can drop a dwarf down to speeds where they take one step per day if they have bulky enough equipment.
Exhaustion is, somewhat, taken into account. Just plain moving doesn't exhaust one, and possibly including exhaustion into just moving if they are sufficiently encumbered is a good idea, but everything that does exhaust a dwarf already does so more when they are encumbered.