In The Corbomite Maneuver, about 7 minutes in, when Kirk enters the bridge from the lift, the camera is tracking him from behind, you see the double doors very clearly. Less than a minute later, a crewman enters and you can briefly see the grey inner door close before the red outer door does.
Tomorrow is Yesterday at 9:28, Kirk and Capt. Christopher enter the turbolift and we see the double doors. In fact they move and sound a like a contemporary elevator door (you can hear the mechanism "jiggle").
The Man Trap: you can see edges of the open inner doors if you look, but we don't see them move.
There aren't many examples and I'm hard pressed to think of any more. Sorry, I can't find any pics to post, TrekCore didn't capture those shots.