Some of them do. (A few of them do it quite a lot.)
Most of them don’t.
This is assuming that the question defines peeing as "expelling liquids from the rear end." Not all insects really do that. Most insects will leave mucky droppings as sole forms of excrement. A few of them, however, can shower you with excreted liquids.
Let me explain: Most insects' excretion systems are contained in their gut, meaning they have no true kidneys. They do have something called Malpighian tubules, which are like fine tentacles shooting off their guts that collect liquid nitrogenous excreta from their "blood" ... into their hindgut. This could be seen as urine that doesn’t really get urinated, because it is released into the gut contents, from which all needed water is reabsorbed. This means all insect crap comes from the same hole, usually in a fixed presentation format.
Caterpillars, for instance, don't pee but they do poop a lot—leaving little black bags around plants. Wasps rub the ground, leaving a mucky brown streak as their sole form of excrement. However, some species do expel liquids. Aphids are famous for eliminating a droplet of “sugary pee” called honeydew that other insects may appreciate sipping from. Cicadas, meanwhile, are famous for peeing like six-legged racehorses.
Have you ever been showered by cicadas in summer? It feels great! Why do they eliminate so much water? Because they take too much of it in, but that’s another story.
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