So, here is what I have observed so far:
I stop the car (REX not running). Press the STOP button a 2nd time (with foot off brake) and most of the interior lights (including the i-Drive screen) turn off. the speedo screen stays on with the BMW "I" logo, as do many other dashboard lights and the interior door handle lights, and the STOP button pulses orange. When I open and close the door more lights go out. When I lock the car the red hazard light button and the dashboard lights surrounding the headlight switch stay lit.
I checked it every 15 minutes, for an hour and 45 minutes, before I gave up and went to bed. Those lights were still lit. However, this morning when I checked it, all lights were out. So, they do go out, but it takes a LONG time. This is a new occurrence for my car, and I suspect that it started a few weeks ago when the dealer replaced the Body Control Module. Maybe newer i3's all act like this and now mine does too with a new BCM? This is all very strange to me as I can think of no beneficial reason for the hazard button light and the lights surrounding the headlight switch to stay lit.
On a related note, in my original post I also mentioned that the fan icon on the temperature control dial stayed lit, even though there was no active our upcoming Departure Time scheduled. Well, there was a Departure Time scheduled but not for 3 days. I turned off the Departure Time schedule and now the fan icon is off, so that explains that part of it. But again, this is a new phenomenon and it didn't do this before the new BCM.
So, I'm wondering if I need to contact the dealer and report this. My CPO warrant runs out in a few weeks so if there is a problem, I'd certainly want it taken care of immediately. But maybe this is how the newer i3's act? I'm hopeful that some of you with newer i3's can comment.