How does the unidentified old woman in Fahrenheit 451 affect Montag's life?

In Fahrenheit 451, the firemen are called to a house on Elm Street when they receive a tip that there may be books hidden in the attic. When they get to the house, they find the owner of the house recalcitrant and resolute; the old woman simply refuses to concede her defeat. She answers defiantly when the firemen question her about the whereabouts of books in her home.

You know where they are or you wouldn't be here.

In the past, the police had always preceded the firemen; they could always be depended upon to usher the inhabitants of the home away before the burning started. In this case, however, some human miscalculation has engendered such a twist in affairs that a rebel has managed to plant herself right in the path of the flamethrowers. As the firemen drench the books with kerosene, Montag desperately begs the old woman to come away with him. However, she refuses. Her resolve mirrors that displayed by two men, charged with heresy during Queen Mary's reign, who died courageously at Oxford on October 16, 1555.

Her courage in the face of death is the catalyst for Montag's own rebellion. He starts to question the wisdom of burning up a defenseless old woman with her books. In his conversation with Mildred, he reasons that the books must represent something of value if they are powerful enough to propel an old woman to die with them. Beatty's later explanation about the need to destroy controversy of any kind and the need to preserve uniformity at all costs fails to satisfy Montag.

We must all be alike. Not everyone born free and equal, as the Constitution says, but everyone made equal. Each man the image of every other; then all are happy, for there are no mountains to make them cower, to judge themselves against. So! A book is a loaded gun in the house next door. Burn it.

Due to the old woman's actions, Montag is no longer satisfied with the status quo. Far from accepting Captain Beatty's words, he finds himself compelled to confront the realities of his life and his work as a fireman. Bear in mind that all revolutions start this way: one man questions and seeks answers and then feels compelled to seek out others of like mind (later, you will see Montag join a group of book rebels). So, you can see what an impression the old woman's courageous death has on Montag.

Hope this helps!