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On mine it turns off after a while (and turns back on as I walk back to the car - tech package with 'open on approach' or whatever it's called).
Apologies: posted without paying attention to the forum this was in (just reading 'New Posts' as I often do). I have a Model S, not a Roadster, so what I said here is completely irrelevant to the question.
Sorry.
Nope, if the car is plugged in the VDS stays on. It bugs me as well for the same reasons the OP mentioned but I have a "newer" Roadster and after almost 3 years no signs of fading.
CRT's and Plasma TV/Screens are the screen technology that is highly effected by burn-in, not LCDs so much (from what I found a very small portion of them do).