UPDATE AND A WARNING:
I took my car out last night (didn't get home until 3:45am) so I had a chance to see my new LED dome lights in action.
1) THEY DO NOT TURN OFF. After they dim down, the LED lights are very clearly still burning, albeit extremely dimly. After shutting the car down I just stayed inside to see if they would turn off, and THEY DO NOT. The LEDs stay lit whether the car is running or not. I assume that the SJB will power them down at some point but I wonder if the current draw would be enough to kill my battery over time.
2) I'M NOT IMPRESSED WITH THE LIGHT. The white LEDs look cool, but in my opinion they do not put out more light than the stock incandescent bulbs. If anything, I feel they put out LESS light. Maybe it's just because the light is bluish white and just isn't as visible, I don't know.
That being said, I'm pulling the LEDs today after work. I can't have lights constantly burning over a long period of time. Falchion, you may want to check yours at night!