One aspect of the character of Job in "The Book of Job" that Montag in Fahrenheit 451 mirrors is Job's quest for meaning.
Job wonders why his God has deserted him, why he is being visited by so much misery. Why him? He is searching for meaning.
Montag's entire mental journey in the novel is a search for meaning. His life doesn't have any, and neither does his society. No meaning is derived from the TV programs, from the near death of his wife on the part of the technicians who save her, from the wars that are so distant, from the lack of real conversation, etc.
Montag, like Job, searches for meaning. His search leads him to wonder if meaning can be found in books, and that is where he ultimately finds it.
As dstuva comments, The Book of Job in the Bible is all about one man trying to process and understand what has happened to him. The character Job suffers immense disaster - the destruction of all of his wealth, the death of his family and children and even personal sickness. He seeks to discover why God caused all of this. What is interesting is that God's answer to Job basically says that God's ways are higher than man's ability to understand and we cannot understand the thoughts of God. Also highly significant is the restoration of Job - he becomes even richer than he was before as is given a new future. Does this hint at a happier future for Montag? And is Job mentioned at this point in the novel because of the intense questioning that Montag is going through?
I think that Faber chooses the Book of Job to read to Montag because that is the book of the Bible that is most closely connected to suffering. Montag is clearly suffering.
To me, Montag is not really the equivalent of Job. Job has way more problems than Montag has. However, Montag does definitely have problems. His major problem is that he does not feel that his life has meaning. He feels that his job is actually a bad thing for society. He feels that his wife does not care about him. And he feels that his whole society is bad.
Because of these things, he feels oppressed, just like Job does.
Job's trials include a major dispute with his wife, bad advice from his friends, and consternation over the meaning of his own existence. Job's wife wants him to curse God - Millie wants Montag to destroy the books. Job's friends have only poor excuses for the cause of his troubles - Beatty's explanations are similarly lame. Job wonders how and why his world has been turned upside down - Montag does essentially the same thing.