This question was inspired by a recent comment to a previous question of mine. In that question, I asked about the literary effect of dropping the articles from titles of novels and films. Even there, I suggested that probably there isn't one reason, or even one group of reasons, that works for all titles. A reasonable conclusion is that one should probably investigate this issue on a case-by-case basis. So, let's tackle Animal Farm.
Here is the comment, by user14111:
If I remember right "Animal Farm" was the new name of the farm after the revolution. If so, that would explain why the title of the book is Animal Farm rather than The Animal Farm. You could then ask why the pigs didn't call it "The Animal Farm" but that would be a different question.
I think this is a very good observation. But here is my reply.
First of all, there is a bit of a 'chicken-and-the-egg' problem here. Yes, the title of the novel is taken from the post-revolutionary name of the farm. But the events on the farm aren't a historical given, constraining Orwell's hand. On the contrary, they are purely fictional, and Orwell can make them be whatever he wants them to be. In fact, given the importance of choosing a good title, I think it is more likely that the causal chain went the other way around: Orwell may well have decided that Animal Farm sounded better as a title than The Animal Farm, and that is why he had the animals rename the farm into Animal Farm rather than The Animal Farm.
But we don't need to solve this chicken-and-the-egg problem (which is good, since we can't go back in time and ask Orwell how he actually proceeded). All we have to do is reformulate the question slightly.
Let's recall that, in the novel, the original, pre-revolutionary name of the farm (to which it reverts near the end of the novel) did include the article: 'The Manor Farm'. Here is how Orwell describes the name change:
Napoleon sent for pots of black and white paint and led the way down to the five-barred gate that gave on to the main road. Then Snowball (for it was Snowball who was best at writing) took a brush between the two knuckles of his trotter, painted out MANOR FARM from the top bar of the gate and in its place painted ANIMAL FARM. This was to be the name of the farm from now onwards.
So, the definite article had been dropped from the sign even while the farm still had the old name. This is surely consistent with the 'language of signs and notices': we often drop the articles in signs. But we also often don't drop them. Orwell could have included the article in the sign to begin with, but he chose not to.
Furthermore, having replaced MANOR FARM with ANIMAL FARM on the sign, the animals could have treated it the same way as the old sign: as a shortened version of the full name, where the latter retained the definite article. But Orwell decided otherwise.
In short, the fact that the new name of the farm was 'Animal Farm' (without any articles) is a consequence of several decisions by Orwell, none of which were forced on him by anything external—they were purely creative, artistic, literary decisions.
What is the literary effect of these decisions of Orwell's?
Here is another way to phrase the question. Imagine that instead of the text I quoted above, the text of the novel instead read,
… painted out THE MANOR FARM from the top bar of the gate and in its place painted THE ANIMAL FARM. This was to be the name of the farm from now onwards.
Imagine further that everywhere in the novel where it says 'Animal Farm', it instead said 'The Animal Farm'—including in the title.
What literary effect would these changes have? How would this hypothetical novel be artistically and literarily different from the actual one?
(Notice that I am not phrasing the question as 'why did Orwell make the choices he did'. Again, we can't go back in time and ask him. And even if we could, he may or may not have a good answer. For all we know, he made these choices simply because 'they sounded better', and that's all he could tell us. But we can still ask what the literary consequences would have been if he'd made different choices.)