Please do not __ this door after midnight.
Why would you not just say "open or close"? It's unambiguous, short, and simple. I don't understand why you think you need a single word.
Please do not open or close this door after midnight.
I like operate or just use.
As an alternative to the other answers:
Please do not touch this door after midnight.
Swinging after midnight not allowed?
I suppose toggle might work.
This door must not be used after midnight.
This door is not to be used after midnight.
Since we seem to be implying there is a detriment to operating/using/toggling the door why not specifically ask the person not to incur the detriment: "Please don't creak the door after midnight!!" Just enumerate the negative effect as the action to avoid.
I'd say swing in that case
What about displace, manipulate, interact with or tamper with?