Hi,
mine did the same, though Audi recently fixed it under warranty. It was the microswitch in the door lock mechanism.
The problem, as well as the driver info display telling me that the door was still open was that the interior lights would stay on too. Gone are the days of a physical 'push-in' switch in the door jams to operate the interior lights. It's all done by computer these days.
More worryingly though, sometimes the car will not allow you to lock it up as it still thinks the door is open.
This was a very intermittent problem for me, but only seemed to happen when the weather was hot. I never thought it would be anything to do with the weather, more to do with the fact that when the weather is hot, I'd have the window open and use the full closure when locking my car.
Since it shows on the driver display, I think that it should show a fault code, that's what I'd been told by Audi anyway.
Good Luck.