Per the
Wikipedia article, common clothes moths lay eggs whose larvae eat just about every kind of fiber, though with a preference for wool. They will eat stored foodstuffs as well, though most forts keep their food in sealed containers, so I doubt that angle would be worth implementing.
The game already has regular moths, that do nothing but fly around as vermin--nothing special about them. But if a certain percentage of those were clothes moths, and you don't have enough cats, they could flutter in and lay their eggs unnoticed. If left unchecked for a couple of years, these moths could spread throughout the fort, causing unhappy thoughts about nearly-new clothing already having holes, and the constant annoyance of the moths themselves . . . essentially a low-grade clothing tantrum spiral. Not only would this provide a new kind of challenge (and a completely non-lethal one at that, another welcome novelty), but the countermeasures will be fun as well: Will you build a full-time Laundry, feverishly boiling every piece of fabric in the fort three times over? Will you dig the dwarven equivalent of a sheep dip? Will your Pressers get to work manufacturing lavender oil? Will you switch to all leather clothing for a couple of years, and weather the liaison's and merchants' complaints about picking up moths at your fort? Will you dig some all-new temporary dormitories and food stockpiles, while you flood the
old fort with water (for a week) or even magma (for an hour)? Will the infestation get so bad that you encourage your dwarves to use hair oils, brushing it into their scalps & beards to deter the pests?
Making (or reworking) other invasive pests, like cockroaches, carpenter ants, etc., would follow a similar template, and be just as much fun to get rid of, but they lack the clothing angle, which I think makes the moths uniquely interesting.