Technically no, but basically yes. I haven't used a func_door with that as it never occured as something I have time for, but I've had an overlay attached to a physbox, which proves it as possible, I'm gonna do it tommorow to see if it works, as it's late at night, i'm tired, and my computer has to update in a minute. So for 'bout 15 hours till I'll have a chance to use sdk.
EDIT: screw what I said, I tested it and it works basically, I just had to select the face of the door, and load the map up, thank god I already had a map I was working on. But it was kinda weird though, as the door was moving, it lagged behind. Might just be my computer, but doesn't look too good, it'd be better to add a thin brush attached to the door with the overlays texture.
EDIT 2: did some calculations, if you just group the door and the overlay together, it should be more realisic when moving. If not, well, I came to the conclusion on my iPhone a minute ago. And I'm signing off for the night. Later.