In the film Avatar we see Jake sleep in his avatar body and wake up in his human one and vice versa.
Does this mean he effectively moves from one body to another and never needs to rest or does he need to spend some time sleeping in his human body?
As far as I'm aware there's nothing in the film which indicates either way (but then there's not a lot about sleeping habits in many other films). Is there anything in the extended canon?
Yes, Avatar drivers do sleep. In one of his videologs, Jake talks to Grace saying: (Emphasis Mine)
STEREO VIDEO-LOG IMAGE --
Jake has just switched on the camera. He looks tired.
JAKE: Do I have to do this? I need some rack.
GRACE, behind him, looks up from her MICROSCOPE, scowling.
GRACE: No -- now, when it’s fresh.
I'm certain this is sufficient proof that Avatar drivers do sleep apart from their Avatar-time.
Source: Avatar Movie transcript
The operator needs to sleep. The issue is that because of the pressures of research, video-blogging and the additional tutoring he's getting on Na'vi society (as well as the time he's spending awake in the avatar body), Jake isn't being allowed to sleep in either body.
Simply put, when the operator is controlling one of the bodies, either their own or the Avatar body, they're awake. When they're asleep in either body, they're asleep.
The official Avatar : Pandorapedia states that when the consciousness is elsewhere, the vacant body is in a sleep-like state. Note that while the body is catatonic, the mental functions are completely intact. The script states that it appears that the operator's facial expression is similar to "a dreamer in REM sleep..."
While the human controller remains in a sleep-like state in a psionic link unit, his or her personality inhabits and completely controls a custom-made Na’vi body.
Pandoripedia
Within the script itself, we see the consequences of this sleep deprivation on Jake:
Jake looks like he’s about to collapse. He’s gaunt, eyes deeply shadowed, hands shaking.
TRUDY : You need to get some rack.
Jake grabs a packet of freeze-dried crystals and pours them directly into his mouth, chewing them up.
JAKE : Gonna have to settle for coffee.
