To me, Montag's main dilemma in this story is how to get the life that he wants to have. How will he be able to have a life in which he can care about other people without getting in trouble with his society?
Almost from the beginning of the story, we see that Montag is not satisfied with the emotional part of his life. He does not feel that he and Milie have a true relationship, for example. He wants that to change -- he wants to truly care for other people and to have real relationships with them (sort of like Clarisse did).
But Montag's society does not really allow for such things. People who try to act like that are liable to get themselves killed the way that Clarisse seems to have. So, to me, this is Montag's dilemma -- how to make life worth living without ending up actually dead.
To answer this question, take a look at the scene after Beatty has explained the history of the fire service to Montag. Alone with his wife, Montag says:
I don't know what it is. I'm so damned unhappy, I'm so mad, and I don't know why I feel like I'm putting on weight. I feel fat. I feel like I've been saving up a lot of things, and don't know what. I might even start reading books.
Arguably, Montag's dilemma is that he is unhappy but does not know why he feels this way or how to solve it. The fact that there are few people to whom he can turn exacerbates the problem. Mildred, for example, is consumed by the parlor walls to the extent that she is neither interested nor concerned. Clarisse has disappeared, and Beatty is a guardian of censorship. Montag, therefore, has nobody to help him solve this dilemma, no one to help him work through his unhappiness and find a suitable solution.
It is not surprising, then, that Montag next turns his attention to Faber, a retired college professor whom he suspects of possessing and reading books. For Montag, Faber is the only person who can honestly answer all of his questions and who can help Montag develop a plan to improve his state of mind.
What forces aid Montag in the resolution of his problem or dilemma? How does Montag resolve his dilemma?