If you don't sleep you cannot dream, when you dream your brain enters a state known as REM sleep, or Rapid Eye Movement, at this time your brain rests and this is what keeps you sane, because your brain needs to rest if you brain doesn't get any REM sleep for long enough you will go mad, madness has a habit of lacking self preservation which includes not recognising the need to eat or drink, doing dangerous things without concern for your own safety, hallucinating, paranoia etc.
However....there is also a point where you will just pass out through exhaustion so staying away is pretty difficult to do unaided.
Personaly I think it would be awesome if slowly you started seeing things that were not there, for example you would see a deathclaw as a friendly npc or you would lose track of your health meter because it goes invisible, your companion would turn into a giant cartoon vault boy. Or your actual faction rep would appear to change so that if Cesars legion hated you they would show up friendly and if NCR likes you they show up as hostile totally screwing with your perception of what is right and wrong. It would never happen because too many people would complain about it but it would be a way of getting your crazy on.
I would love this, but make it so you have to have wild wastelander trait so it can be avoided. This would be hilarious. you walk up to a "Friendly" Deathclaw and try to talk to it. It starts saying, "O, hey... You look yummy! O, mommy wants me.... will play some other time" Then it would disapear. Or like people you meet in the game, like benny.
That would be hilarious! Your looking for Benny then you see him and pull out you knife. You run at him and swing, then he vanishes. That would make hardcoe so fun, o and somtimes your screen will dim, the more tired you are the more it dims, and the worst your perception is. Those with a high perception won't get tired often, those with a low perception would need sleep alot.