It might be more effective to compare a specific Shakespearean play to Fahrenheit 451, but in general, Bradbury and the bard do share quite a few similarities.
1. First, both authors rely heavily on Biblical allusions, and in Fahrenheit, the preservation of the Bible and Shakespeare's writings represent part of Montag's rebellion against his conformist society.
2. Montag's relationship with his wife, Mildred, mirrors some Shakespearean relationships--most notably, Macbeth and Lady Macbeth's marriage. Mildred wants her husband to be more ambitious as does Lady Macbeth. Both women are deeply troubled and view suicide as a means of escaping their troubles, but their husbands are helpless when it comes to helping their wives. Ophelia from Hamlet also shares similarities with Mildred.
3. Captain Beatty, like Iago with Othello, manipulates Montag and causes him to question what he believes and why he holds those beliefs.
4. Finally, many would view Montag's journey as a tragic downfall (at least in the eyes of his family and superiors). At the novel's beginning, he believes he is content in his life, has a steady job, and seems to be respected at work; yet his intellectual curiosity (his tragic flaw) eventually leads to his losing everything but his life. The primary difference between him and a character like Shylock fromThe Merchant of Veniceis that by losing everything, Montag is actually happier and demonstrates hope for his future. His downfall is not so "tragic" after all.