One way in which Montag is heroic in Bradbury's Fahrenheit 451 is the way he tries to save the woman at 11 No. Elm, a house the firemen are sent to burn because the owner has been harboring books.
When they arrive, Montag is startled because they never have had to face the owners: they had already been removed by the police when the firemen arrived. In facing the woman whose house they are prepared to burn, it feels much more personal to Montag. Before, he felt as if they were only destroying unfeeling things. Now he has a sense that they are invading a person's home, destroying things that mean a great deal. Even though the woman is scornful of Montag, he still tries to get her to leave the house.
"Come on, woman!"
The woman knelt among the books, touching the drenched leather and cardboard, reading the gilt titles with her fingers while her eyes accused Montag...
"The whole house is going up," said Beatty.
The men...glanced back at Montag, who stood hear the woman.
"You're not leaving her here?" he protested.
Beatty doesn't try to save the woman. But Montag reaches out to her, pleading that she leave with him.
"You can come with me."
"No," she said. "Thank you anyway."
There is a heroism in this action. Prior to this moment, Montag believed he was doing his duty as prescribed by society. However, once he is presented with the reality of the situation, with the humanity in the equation, he changes. He does not dismiss the situation as Beatty and the other firemen do. He tries to at least save the woman from dying. Ultimately, she ignites the fire that burns her house, her books and herself, but Montag is a different man.
Later in the novel, Beatty taunts Montag at the firehouse just before they are sent out on another call. Montag believes it's just another job, and is certain he cannot force himself to burn one more home after all he has witnessed and learned. Surprisingly, the truck pulls up at his own house, while Mildred rushes past him to get in her car and leave: she has turned him in.
So Montag helps burn his house. In doing so, he disconnects himself from society and all it stands for: the brain-numbing Seashell radios, the mind-controlling TVs, and his mindless existence. Then Beatty tries to arrest Montag, hitting him. The green bullet Montag has in his ear as he listens to Faber's directions, falls onto the ground. When Beatty sees it, he realizes that Montag is working with another "criminal" and he threatens to hunt Faber down. It is at this point that Montag rebels—he is called upon again to try to save someone else. He takes a stand against Beatty.
"No!" said Montag.
He switched the safety catch on the flamethrower. Beatty glanced instantly at Montag's fingers and his eyes widened the faintest bit.
Beatty shows no sign of retreating and Montag kills him. At this moment, he is committed to a new course. He destroys the Mechanical Hound that tries to kill him. The government sends every resource available—even calling out Montag's neighbors—to catch him. These steps drive Montag further away from the physical, mental and social captivity exerted upon him by the government.
Beatty once said quoted Samuel Johnson:
He is no wise man who will quit a certainty for an uncertainty.
Before, an easy life was certain for Montag. Breaking the law created uncertainty. Montag's willingness to fight society may not have been wise, but it made him a hero.