This has been discussed ad nauseum in the past.
Turning it off spins down the hard drives -- however, they spin up once an hour to do housekeeping, and to make scheduled recordings. People who are light sleepers with a unit in their bedroom sometimes have problems with the noise of spinup/spindown. When you turn it back on it takes several seconds to spin up before you get picture/sound -- very annoying to me.
Leaving it on is fine; the disks keep spinning and so when you turn the TV on you get instant picture. This is nice too, because the buffer is always there. If you leave it in "live tv" mode, when you turn off the TV, when you wake up at 6:35 you can rewind to 6:30 and see the start of the show. However, the A/V outputs are still active. This can cause problems if your system is more than your TV. I have mine hooked through my sound system as well as my TV. After the second or third time I woke up to hear "*MASH*" blasting through the speakers at 3:00 AM (someone turned the TV off but left the receiver on -- when a recording started BOOM! sound to the speakers), leaving it on wasn't an option.
I compromised between the two. I "disabled disk spindown" so that when I turn the unit off, I don't shut down the hard drive. The advantage is that I don't have the constant starts and stops many finf annoying, and I have "instant on". One of the ReplayTV guys explained that when you "disable disk spindown", the on/off switch just turns the A/V outputs on and off.
Do some searches on "clawfoot portal" and "disable disk spindown" for more info; or find a link to the FAQ (it is gone from the top of the page but in several people's sigs.
Joe