In Fahrenheit 451, the conflict is presented as Montag's increasing awareness of the suppressed state of the society he's been living in. Prior to his conversations with Clarisse, Montag never questioned the act of burning books. He passively went about his life, taking pride in his work without thinking twice about it. However, after Clarisse sparks his imagination, he begins to wonder about his own happiness and he eventually desires literature and knowledge. The initial conflict emerges with Montag. He wants to read and to learn but he lives in a world where that is discouraged. The plot follows Montag's awakening and then broadens to include other characters he comes into contact with. Montag's inner conflict drives the plot.
After Montag's awakening, he elicits the help of Faber. He reads poetry aloud to Mildred and her friends. He is eventually caught by Beatty and the firemen. Montag has no choice but to run away where he finds Granger and the book people. In short, Montag's awakening caused his inner conflict. As he attempts to feed his new thirst for knowledge, he affects those around him, and this includes his attempt to get his wife to see the inspiration of reading. So, Montag's inner conflict becomes a social conflict, culminating in his confrontation with Beatty and his eventual escape.
Meanwhile, there is an additional conflict, a war, developing as Montag escapes.
Since conflict is an integral part of plot, the conflicts in Fahrenheit 451 are what unfold characterization and develop themes, as well as what moves the sequence of events forward to the conclusion.
The initial conflict of the main character, Guy Montag, is an internal one which is sparked by the inciting incident of the plot: his encounter with the pedestrian Clarisse--"a strange meeting on a strange night." Clarisse disturbs Montag's complacency and apathy (a theme) by asking him the question, "Are you happy?" Disturbed by her question, Montag begins to examine his life with more objectivity, especially after he enters his home and finds his wife near death from having ingested too many sleeping pills.
After Montag seeks emergency help for Mildred and her life is saved, he is so moved by his experience of witnessing a woman who loved her books so much she chose to burn along with them that he tries to communicate his feelings to Mildred and awaken her spirit, as well. However, the apathetic Mildred becomes alienated from Montag. With this theme of Alienation developing from conflicts between Montag and his wife, Montag begins his journey into the enlightenment of his mind and the exercise of his free will and human emotions. He is aided by Professor Faber who guides him in his passage to both intellectual and physical freedom.
The plot reaches its climax when Montag sets fire to Beatty in order to escape after his own wife has informed on him as a holder of books. With the assistance of Faber, Montag finds his way to the secret community of readers who have memorized books in order to preserve them. Thus, the conflict of Man vs. Society is resolved when Montag escapes from the Hound, and the government kills another man to cover the error of having lost Montag. The new war that has begun distracts the public from the story of Montag and he is then forgotten.
Clearly, then, the internal and external conflicts have developed the themes and propelled the plot of Bradbury's disturbing novel to its conclusion.