The only thing in the optical viewfinder that would be illuminated would be the focus points and the display along the bottom that provides exposure and metering display. The screen itself isn't lit by anything other than the natural light bouncing off the main reflex mirror.
Batteries will "drain" for a few reasons...
1) There is a "clock" battery inside the camera and some some models this is a dedicated coin battery... on other cameras it's a permanent battery but it draws it's charge from the main battery. When the battery is in the camera, it's charging that "clock" battery. This is a very tiny drain.
2) The camera doesn't technically power "off" when you turn off the power switch. The power switch on a Canon is a soft-switch and it's technically in a kind of extended sleep state. But again... this is a very tiny drain.
3) On the 6D, the GPS can keep running even if the camera is powered off. If you enable the GPS then it stays running at all times, unless you (a) remove the battery or (b) use the menu to switch the GPS off. For this reason, 6D owners should either remember to do one or the other or they'll put their camera away for the night, wake up the next day and discover they have a dead battery. Canon did this deliberately (I question their wisdom) because it can take a GPS a couple of minutes to establish a position fix. Canon was afraid that owners would switch the camera off momentarily (e.g. to change lenses, change memory cards, or just to set the camera down for a minute.) This would clear out the GPS's position and when they flip it back on the camera would have no GPS position and would need another couple of minutes to re-acquire a position fix. SO.... they decided to keep the GPS running EVEN if you power down the camera with the main battery switch. Owners have acknowledged understand the logic of not wanting to wait a couple of minutes just because you powered down long enough to switch lenses or memory cards... but asked Canon to provide a firmware update with a time-out (e.g. if I power up the camera in a minute... I was swapping lenses or memory cards... if I still haven't powered back up after 15 minutes... I've probably put the camera away and they should switch off the GPS.) Unfortunately Canon has come out with no such firmware update.
4) Batteries will (and do) "self discharge". They will do this EVEN if you remove the battery and let it sit on the shelf.
I have a camera body I hardly ever use and after it's been sitting unused for long enough, I can pick it up and notice that the batteries have drained down a tick. I have to wait a few months to notice to notice this, but it will eventually happen if I wait long enough. It's no big deal. I have lots of batteries and I use battery grips on my two 5D bodies so I have a ton of battery power to spare. I have NEVER come even remotely close to running out of battery power even with a full day of heavy shooting. But if that ever did happen and I didn't have a spare to swap in... the battery grip has a "AA" size battery tray that slide in to power the camera while I recharge the other batteries AND I have the AC adapter for when I'm working indoors (the AC adapter is really only used when I'm imaging at the observatory because the really long exposure times will kill a battery much faster than a traditional exposure.)