The climax of Fahrenheit 451 is when Beatty tries to burn Montag’s house down and he turns a flamethrower on him.
Montag is a fireman. In his world, that means that he sets fires instead of putting them out. Houses are fireproof, and Montag and the other firemen burn people’s books. Books are against the law, and so is sitting and talking, and even doing anything slowly. People go fast and watch TV.
When the old woman decides to die along with her books, Beatty scoffs at her.
"Where's your common sense? None of those books agree with each other. You've been locked up here for years with a regular damned Tower of Babel. Snap out of it! The people in those books never lived. Come on now! " (part 1)
Yet Montag realizes he was wrong. When Montag finds out what books really are, he is fascinated. His wife says they are not real, but to him the television shows are not real. The books have real emotions. They are not cold and stale.
Montag gets angry when Beatty turns on him. Beatty seems to relish the moment.
A problem gets too burdensome, then into the furnace with it. Now, Montag, you're a burden. And fire will lift you off my shoulders, clean, quick, sure; nothing to rot later. Antibiotic, aesthetic, practical." (part 2)
He tells Montag he needs to burn his own house down, with a flamethrower, because it was his mess and he needs to clean it up. Montag is so angry that he turns the flamethrower on Beatty, killing him.
The climax of a book is its turning point. The flamethrower incident is the climax of this book because it is when Montag completely turns his back on his old ways and goes on the run.
From this point on, Montag is a different person and the book has a different problem. He has to deal with the consequences of his actions, and find a way to save the books.