There are no explicit rules that require this style of play, and there is a substantial body of rules that are implicitly contradictory. (There's nothing explicitly against it because it's so contrary to the history of how roleplaying games are played that no designer has ever needed to say "don't do this." There is also a lack of rules about not eating the books, for similar reasons.)
What follows is merely a modest sample of the rules from 3.5 and both editions of AD&D that would be incompatible with or nonsensical in the context of not being able to see through doors as the default, "core rules" way of playing.
D&D 3.5 Player's Handbook
Because the designers assumed they'd never have to explain the concept of "seeing", there are no explicity "what can I see rules". We can extrapolate their assumptions that you can see what makes sense to see from how other rules take this for granted.
Surprise: Determining Awareness (p. 137)
Some example situations:
- The party comes to a door in a dungeon. The DM knows that the displacer beasts beyond the door hear the party. Lidda listens at the door, hears gutteral snarling, and warns the rest of the party. Tordek breaks the door open. Both sides are aware; neither is surprised. The characters and displacer beasts make initiative checks, and the battle begins.
Note that the game assumes that PCs can gain awareness of creatures beyond a door via their senses. It would be nonsensical to say that you can only hear the enemy through a closed door, but can't see them through an open one.
Line of Sight (p. 139)
The only explanation of line of sight is by a diagram. The diagram shows that only solid objects such as walls prevent something from being visible to the PCs. An open doorway is not a wall. The fact that such a crucial part of the combat rules is not given an technical definition means that the common English meaning of the words "line of sight" is in force. (Unless we are going to discard thousands of years of how language works, which would be silly in a game that's mostly reading and talking.)
Concealment (p. 152)
Besides cover, another way to avoid attacks is to make it hard for opponents to know where you are. Concealment encompasses all circumstances where nothing physically blocks a blow or shot…
(Emphasis mine.) We all know that an open doorway isn't a physical barrier (as required for concealment to apply), which would be nonsense if you couldn't see an opponent through a doorway.
Tactical Movement: Hampered Movement (p. 163)
… poor visibility can hamper movement. When movement is hampered, each square moved into usually counts as two squares.
Open doors don't hamper movement in any way, so it doesn't make sense for them to limit visibility either, let alone completely block visibility.
Exploration: Vision and Light (p. 164)
In an area of bright light, all characters can see clearly. A creature can't hide in an area of bright light unless it is invisible or has cover.
This section is the entirety of the rules on visibility when cover and invisibility aren't relevant. Therefore given light and an open door, PCs can see into a room. No rule exists in the section (or elsewhere in the game) that says open doors have a special "can't see through" property.
One of the options explicitly listed as a possible action the party can take if they win initiative is (DMG p. 61):
Avoid engagement (flee, slam door, use magic to escape, etc.) if possible.
Slamming the door to avoid engagement would be nonsensical if they couldn't see through the open door. Furthermore, it indicates that "combat" has already begun (initiative has been rolled) before the party enters the room.
The following quote is in the context of the distance at which encounters begin.
Line of sight (DMG p. 62)
If this is unobstructed and light is involved, the distance possible for determination of another party present is virtually infinite. It could likewise be sharply restricted due to obstructions.
An open door doesn't count as an obstruction for any other rule, and there is no indication that vision is a special case. None of the other factors that go into determining encounter distance include "stepping through the door", though the text is otherwise exhaustive, mentioning various magics such as teleportation and invisibility, lighting, surprise, tight spaces, and noise. One would think that Gary would have mentioned the power of door thresholds if it were indeed part of the game as-intended.
AD&D 2nd Edition
The only limitations on vision in the Player's Handbook are distance (Limits of Vision, p. 117) and lighting conditions (Light, p. 118). Doors aren't mentioned, but as when open they block neither distance (that would be an interesting space-time trick) nor light, it follows that they don't limit visibility.
Chapter 13: "Vision and Light" of the Dungeon Master's Guide is dedicated to, naturally, vision and light. It begins with (emphasis mine):
The ability of your player characters to see something and their ability to be seen are
important to the play of the AD&D® game. Characters unable to see monsters have a nasty tendency to be surprised. Characters stomping through the woods waving torches tend to give away their position, making it hard to surprise others. For these and other reasons, you should always be conscious of visibility and light sources when running an adventure.
Nothing that follows mentions that open doors prevent this "important to play" ability. The second emphasised phrase also places the limitations of sight within the scope of "visibility and light sources", neither of which are impeded by open doors in the rules or in common sense.
In summary, these samplings of rules draw a very neat, clear trend line that says open doors don't block line of sight – and is in fact incompatible with many of the actual rules – in any edition of (A)D&D your DM has said they play "by the core rules." I'd suggest throwing the book at them, in the figurative sense, or just leaving the group. A DM that has no players (and who is unlikely to get new ones, at this rate) is more likely to consider "why", and is in the meantime not doing any harm to the limited leisure time of their former players.