At the beginning of Ray Bradbury's Fahrenheit 451, Montag is a good citizen who carries out his job without question, burning down homes of book owners. It is not until Clarisse McClellan begins to ask him questions about the world and what he thinks that he is unable to live his life at face value: he loses the ability to move unquestioningly day-to-day, and begins to challenge (something the government does not condone) society's rules, and what he is doing with his life. When Clarisse asks him what he thinks, Montag realizes that he doesn't!
Several other things take place to change Montag's view of the world and his behavior. Perhaps the most impactful experience he has is when he and the other firemen go to burn a house. However, the owner—an older woman—not only refuses to leave, but also strikes the match that destroys everything, including herself. As Beatty counts, the woman speaks:
"You can stop counting," she said. She opened the fingers of one hand slightly and in the palm of the hand was a single slender object.
An ordinary kitchen match.
Refusing to let Montag save her, he is horrified as the woman ignites everything by her own hand. Montag is devastated because before, the fires were impersonal to him: he never saw the owners who were always gone before they arrived. This, then, makes what he does personal.
What was a casual interest becomes nearly an obsession: Montag is unable to stop himself, and begins to steal books and read them. He makes the acquaintance of Faber, a man who secretly values books just as Montag does. Faber also questions society. Montag reads to Mildred's friends who are horrified. He challenges society by asking questions of Beatty and what they do. Montag's concern focuses on society's attempt to control people's minds. He comes to see the inherent injustice of desensitizing human beings and manipulating their perceptions and actions.
In the end, Montag's wife turns him in to the authorities for having books. Beatty torments him as they arrive to burn Montag's house, and Montag kills Beatty. Only slightly wounded by the Mechanical Hound (which he kills—symbolic of fighting the system), Montag escapes into the woods and across the stream, following others who wish to preserve books, or at least their contents.
Montag is no longer a follower, but a leader at the front of the column of those heading toward a new life:
Montag began walking and...found that the others had fallen in behind him, going north. He was surprised, and moved aside to let Granger pass, but Granger...nodded him on. Montag went ahead.
A hero is...
...a man of distinguished courage or ability, admired for his brave deeds and noble qualities.
Montag is a hero according to this definition. He tries to save the woman in the house. It also takes courage to have books in his society: if caught, he can lose everything, including his freedom—perhaps even his life. He confronts the danger of reaching out to Faber because he wants to learn more. His nobility is seen in his refusal to live in fear; for his desire to change the world for himself and others by questioning society's laws; and, for committing himself (after society is destroyed) to rebuild the world as it once was: with books that not only enlighten, but inspire and entertain.
Montag is heroic: he does not continue to do what he believes is wrong, but chooses to risk everything to make the world a better place.