This is an '05 A3 with a 2.0 TDI BKD engine, about 120k miles on it. The sound starts when I move the key to the ON position, without starting the engine. But even with the engine on, it's still there, and it stops about 10-20 seconds after I remove the key. What could cause this? Here's a video of the sound
That's the anti-shudder valve (ASV), D, in the image below (from an article on how to delete the EGR/ASV – not a good idea on a diesel as the ASV provides some measure of protection against a runaway) it should be fixed open (and I would think quiet) when the car is running. This video "Noise of a Anti Shudder Valve / Inlet Manifold Flap ready to blow" is similar and suggests impending doom (or at least failure of the valve). If the valve fails open, you probably wouldn't notice anything by the quiet (barring a runaway), but if it failed closed then you wouldn't be able to start the engine (or at the very least power would be severely limited).