The app is very limited on what it reports. It simply reports that the vehicle is charging or not charging. As a side-note, I would love to be able to change the scheduled charging time from the app - this is on my wish list. At the moment it's only possible from the main screen.
The Tesla main screen (Charging) shows (on the left) the charging status (e.g. 0A, #miles gained) and on the right it reports the connection status and how much it's drawing in real-time. Its here that I see a very small draw during non-charging times.
I can't say what happens when I leave the vehicle, since the screen turns off. My vehicle is currently being serviced, but when I get it back, I plan to:
1.) Calculate how many kwh is lost per hour and figure out what the amp draw would be.
2.) Determine if there is any vampire drain while the vehicle is asleep and plugged in
3.) Determine the vampire drain while the vehicle is asleep and not plugged in
4.) Try to keep the vehicle "awake" and determine the vampire drain while plugged in. Maybe leaving the key in the car or leaving a door open might keep it "awake".