Schrödinger’s door?
Rather than “on” and “off,” how about if we consider the value of the door to be “high” or “low?” This might correspond to logic states in a digital signal.
If we do, I consider an open door to be “high.”
But the default state is no doorway or door, it’s a wall. So the purpose of the door/doorway is to allow passage when needed. If it wasn’t there, then no access is possible.
A doorway with a closed door is no different from a wall, and therefore, the purpose of a door is to turn a doorway into a wall, restoring the default state.
I’ve heard people (non-native English speakers) say “open the light” instead of “turn on the light”. I think it’s a direct translation of how they phrase it in Chinese and possibly other Asian languages.
I’ve heard that a number of times from people whose first language was not English. As far as I know that made sense in their first language. In any language is a door considered ‘on’ or ‘off’ based on it’s state of being open or closed?
My parents, both born and raised in the Bronx, would also sometimes say “open” (or “close”) the light.
Did they ever say turn the door off so I don’t have to pay for air conditioning all of Woodland Heights!
For what it’s worth, as part of smartifying my house, I set up a few open/close sensors on doors in my house. They use hall effect sensors connected to pins on a micro-controller. The sensor reads low (off) when the door is closed, and high (on) when it is open.
If this were true, we wouldn’t be having this discussion. A door gives the possibility of passage, and therefore is different from a wall. 
No, a doorway gives the possibility of passage, while a door cancels it. Topologically, a closed door is indistinguishable from a wall.
If the door closing mechanism was a force field, it would be off when you could pass through, and on when it blocked the doorway. Same if it was a wall of flames.
So I’m going with “closed door is on, open door is off”
Reading the various responses, I think there may be an underlying assumption that a few replies have touched on but that hasn’t really been considered directly.
In the question, are we considering the door all by itself, or are we considering the door system including the doorway in whch the door is mounted?
In the latter scenario, as a few people have said, a wall without the door system is simply a wall, with no means of ingress or egress. With the door system in place, the closed door mimics the wall, and the wall plus the door functions the same as the wall alone. Therefore, we may consider the default state to be unpassable, and the active state to be open, allowing passage. Consequently, the door is “on” when it’s open, as it directly changes the state of the wall without the door.
But in the former conceptualization, the door is independent of the doorway. The wall and the opening together become something of a system — a passable wall. When the door is hung in the doorway and left open, it’s not doing anything. In other words, opposite to the above, the wall-doorway system has a default state of being passable. And when the door is closed, it is “activated” to serve its function of closing the opening and making the wall unpassable.
In my reply above, I instinctively felt like the wall/doorway system plus a separate door was the right way to consider the question. Other people start with the opposite assumption, considering the wall by itself as one entity and the door plus its opening as the addition to the wall.
I find this interesting. I don’t know what it means, if it means anything at all, but it’s interesting.
A lot of fire doors operate by keeping the door open with an electromagnet - and when the power goes out, the door closes automatically. In that system, open is definitely “on”.
Let us now consider a door with a cat flap.
In my understanding, a door is what a cat wishes to be on the other side of. Whether it’s worth expecting a cat to use the terminology “on” or “off” is another matter.
To a cat, the two states are “boring” and “annoying”. A closed door is annoying; an open door is boring.
Consider portal technology: every wall is a potential door. No portal would be the default, or off value. A portal existing would be the on value.
This is even more apparent when you consider trap doors/portals in floors and ceilings.
If the cat flap can be locked shut, are we getting recursive?
What if the cat flap has a mouse flap in it?