The war hangs over Montag's head throughout the novel; he is not sure who is at war with who, and is confused and irritated that no one else seems concerned. As the novel progresses, snatches of reporting from radios show how the war is moving forward, and as Montag flees the city, the war is officially declared. It is interesting that the war has a long buildup, but the technology of the time allows the war to be extremely fast:
And the war began and ended in that instant.
Later, the men around Montag could not say if they had really seen anything. Perhaps the merest flourish of light and motion in the sky.
(Bradbury, Fahrenheit 451, Google Books)
Montag and the other men at the train tracks are astonished by the power of the jets and bombs; the city is utterly destroyed. For the book hobos, the war is an inevitable part of the societal degeneration that resulted from censorship and collectivism; for Montag, the war is the end of his previous life and the beginning of a new life. He knows now that his unique knowledge, his value as an individual, is important to rebuilding society; all the mindless drones in the city succumbed to the war because they couldn't think for themselves, but Montag has escaped precisely because he rejected those ideals.
When Fahrenheit 451 begins, the threat of war hangs over the country and its citizens. When Montag is at the firehouse in Part One, for example, there is an important announcement on the radio:
A radio hummed somewhere. ". . . war may be declared any hour. This country stands ready to defend its--"
This announcement is significant because it occurs around the time that Montag feels threatened by the Mechanical Hound. It is as though the prospect of war in his society reflects the conflict in his mind regarding his own happiness and his commitment to being a fireman.
The threat of war continues as the novel progresses. For Mrs. Phelps and Mrs. Bowles, the war is not a time to worry: they anticipate a quick result and believe that their husbands will not die in the fighting since they have never known anybody die in a war. Montag is shocked by their total lack of understanding and this, perhaps, prompts him to read the poem "Dover Beach," a poem that details the harsh realities of war and leaves Mrs. Phelps sobbing "uncontrollably."
It is fitting that war is finally declared at the end of the novel when Montag has agreed to stay with Granger and the other professors. For them, the destruction of the war offers the possibility of a new beginning: an opportunity to rebuild society.