Montag is a hero in the book Fahrenheit 451 by Ray Bradbury in that, ultimately, he refuses to conform to the dictates of the society around him - a society that is beholden to the directives of the totalitarian government that rules over them.
At one time, Montag blindly let the ways of the society and culture around him be his guide. However, Montag has a conscience and a sense of right and wrong - a moral compass. He begins to increasingly think for himself and for what he truly believes. This is his first heroic act of any real consequence. He has made a conscious choice to think beyond what the totalitarian regime tells him and others to think.
He is a hero in taking action that is opposed to the regime. He takes books - which are banned - home. He is heroic in that he hides them at home so that he can delve into them later to learn what they are about and how they can help him be a critical thinker, and thus, be a creative and thoughtful human being.
He is also heroic in that he strives to be a 'changer', one who is trying to better society by pointing out its wrong ways. He does not want men to be automatons that do not think for themselves. He is heroic in trying to help his wife come out of her cocoon of fantasy.
He is also heroic in going to Faber and also in seeing the good character qualities in Clarisse. In even thinking that Clarisse is right is a heroic act for Monatg, because he is courageously facing the truth and admitting his true feelings. He knows he cannot continue 'conforming' to a way that he knows and believes is wrong.
Fundamentally, Montag is heroic because he is willing to change. He will not toe the party line anymore. He will pursue what he believes is right with others of the same thinking. He knows the risks in doing this, but is willing to do it anyway.
Montag is heroic in that he reacts against a controlling society that prohibits individuality and imagination. He finds such mindless conformity damaging to the spirit.
Montag rebels against his dehumanizing society in the hope of becoming a real person, one who truly thinks and acts without drugs. Having witnessed the deterioration of his wife Mildred, who has become no more than the visions on the parlor screens, a mere automaton who laughs on cue, and who listens to the buds in her ears, never hearing her heart or that of her husband, Montag makes supreme efforts to re-engage her in life, but she insists upon dwelling in the artificial world of her society. In fact, she is so insistent about choosing this world that she reports her husband to the authorities. But, he defies these powers of society, setting fire to Beatty and fleeing to Faber, who connects him to the people who dwell in another place where they preserve the texts of famous works, the recordings of the thoughts of philosophers, scientists, theologians, satirists, and historians of the past.
While Montag hides from the authorities, he talks with Granger, who tells him about his grandfather, a sculptor, who created things. "He shaped the world. He did things to the world," Granger reflects.
"Everyone must leave something behind when he dies, my grandfather said....Something your hand touches some way so your soul has somewhere to go when you die....It doesn't matter what you do ...so long as you change something from the way it was before you touched it."
Montag changes his society by rebelling against it, by burning what is evil rather than what is good, and he will leave something behind as he has memorized the Book of Ecclesiastes.
Montag transforms from blind civil servant - the "fireman" who operates blindly to the directions of society to as individual who's every action is an act of civil disobedience. The act of possessing a book, taking a book from a fire, talking to the professor, reading the books and fleeing the city made him a criminal. The reader sees these acts as heroic because he is finding his "humanity". It is the act of disobedience and turning away from the societal norms -- finding something valuable in the pages of something forbidden that makes him a hero.
Montag does not see himself as heroic but in memorizing the book of Ecclesiastes so as to save a piece of the past for future generations is brave and heroic act.