I have committed the crime of sleeping in plate armor. Well, mostly plate - I used 16-gauge steel for most of it, but had a skirt of 18-gauge scales over light leather under the 16-gauge breastplate, and a padded underlayer. Didn't wear the helm or the gorget. I was just so tired I lay down for a minute and woke up an hour later.
When I awoke, I was stiff and sore. You see, as I slept, I moved a little. The edges of the scales left lovely little lines in my flesh even through padding. The less said the better about the sides of my limbs, where there was little padding between the steel plate and the flesh. And somehow I had managed to pinch a bit of my flabby stomach between the breastplate and backplate, which left a nice purple bruise for a week or so. And my arms and legs were so stiff. You see, what I worked out at the time, as you sleep and move, your joints move too. The armor doesn't, not the same, so the joints tend to be held in uncomfortable positions, leading to muscle strain and stiffness. Your body is flexible, most armor isn't. And I only slept for an hour (perhaps a little more). Imagine a full night of it!
Now, that said, if a person slept in padded armor or *light* leather (not thick or hardened leather as used in most leather armors worthy of the title "armor"), there shouldn't be any ill-effects except the smell and a reduced lifespan of the armor. Mail armor is going to dig in and pinch, even through padding. Remember "The Princess and the Pea"?
Plate armors are going to be terrible to sleep in. Even if you can, you're going to be stiff, sore, not particularly good in a fight due to those let alone the effects of being not-even-vaguely-well-rested.
In my opinion (and it is only my opinion), people would tend to sleep out of armor. Maybe in padding, if they are worried. If they wake and need to fight, they'd throw on a mail shirt or something similar, something that can be donned alone and quickly, secured by tightening a belt. Possibly a plate cuirass, if it fitted well. Donning a mail hauberk would take longer, but also be relatively quick. Full plate armor? Well, there was a reason for squires. Several reasons in fact, but donning plate armor was definitely one of them, especially if you wanted to be involved in the fight in the first few minutes.