What does the door symbolize in Fahrenheit 451?
The door to someone's home is the gateway which separates the privacy of the home and the public world outside. Presumably, once a person or family is inside the home with the door closed, they are free to do as they please. In that domestic space, they are not under the watchful eye of the State.
However, in Montag's world, the door does none of these things. The Mechanical Hound can sniff out a criminal (book owner) regardless of any door. A door, in Montag's world, does not offer the privacy that it should. When Montag and Millie are looking through the books, the narration is as follows: "They turned to stare at the door and the books toppled everywhere, everywhere in heaps." It is as if the door is not even there to prevent Beatty from getting in. Beatty leaves but only in order to give Montag one last chance to get rid of the books. The door is more like a prop of a fake domestic building meant to seem like an actual home.
For the family in the home, the door is supposed to provide privacy and the freedom to do, read, think whatever they choose. This is not the case in the novel. In fact, in the end, doors are actually used indirectly to help the officials catch Montag:
"Police suggest entire population in the Elm Terrace area do as follows: Everyone in every house in every street open a front or rear door or look from the windows. The fugitive cannot escape if everyone in the next minute looks from his house. Ready!"
Montag imagines thousands of doors opening, people peering out to help the authorities track him down.
So, the door symbolizes the illusion of privacy, the illusion of freedom from surveillance. And given that most people are slaves to the parlour/television walls in their homes, for them, the door represents a kind of imprisonment. Only when the masses are summoned by authorities, via their futuristic television, do they open their doors. And again, they only do so in order to help enforce state law. In this novel, the door is like a jail cell door. It is a door which provides no privacy, no freedom from State control and surveillance. It is the illusion of an actual door.