lots of different things can cause this. this sounds counter-intuitive, but sometimes padded gloves cause this. this happens when the padding is too thick, especially if the rider has small hands. the net effect is that the combination of the bar, grip, and gloves means you're squeezing something that's too big around, sorta like palming a basketball for 45 minutes straight. ouch. hop on and give it a go without gloves. if that works, try a set of gloves with thinner pads, in particular gloves without gel padding (they're the thickest). also, you may want to dispense with bicycle shop gloves. i just picked up a pair of sweet fox motocross gloves at cycle gear for $15. they're no different than a pair of mountain bike gloves, but if i had purchased something similar at a snobby bicycle shop, yeah, they'd be $35. just make sure they have a soft spot to wipe your nose on. since that fix is the cheapest fix (testing it is free), try it first. if that doesn't work, don't worry; there are other solutions, they're just more expensive.
you know, it's funny. i worked in bicycle shops for wwwwwaaaaaaaaayyyyyyy too long and i've been preaching about this the whole time: the accessories are always more important than the bike. but nobody ever listened to me (not that you're one of them). well, a few people did, but that's it. the wrong combination of gloves, helmet, shoes, and shorts leaves riders so uncomfortable that they don't ride. and why would they? it hurts. so peeps would come in my shop and buy $2k rides and $20 shorts and think it's the most miserable sport ever. but the folks that spent $300 on accessories and $600 on a bike ride for years. it's a crazy, self-defeating sport sometimes.
but if you get it all right, it's a blast, especially if you can find a lonely dirt road and get away from all the noise the world has to offer. when all you can hear is your own breathing, the chain, and the crunch of dirt under your tires, well, that's therapy right there.