I believe that the answer to this question is Icarus, the son of Daedalus. He was the guy in Greek mythology who made the wings out of wax and flew too close to the sun. The wax melted, the feathers fell out, and he dropped to earth and died.
Beatty compares Montag to him at the start of Part III. He does this just as they have gotten to Montag's house and Montag has realized what has happened. Beatty tells Montag that Montag essentially got too bold and didn't listen when Beatty tried to warn him.
He says Montag has flown too close to the sun and burnt his wings.
At the beginning of Part Three, Beatty says to Montag, "Old Montag wanted to fly near the sun and now that he's burnt his damn wings, he wonders why" (Bradbury 107). Beatty is comparing Montag to Icarus who was a character in Greek mythology. In Greek mythology, Icarus' father, Daedalus, fashioned two pairs of wings out of feathers, wax, and wooden frames to escape from King Minos' labyrinth. Daedalus cautioned his son not to fly too close to the sun because the heat would melt the wax on the wings. However, Icarus did not follow his father's directions and flew too close to the sun. The wax on his wings melted, and he fell to his death. Similar to Icarus, Montag attempted to go rogue and challenge the authoritative government but was caught. Beatty recognizes that Montag attempted a near impossible feat and was unsuccessful. Fortunately, Montag takes matters into his own hands and kills Beatty using a flamethrower to avoid being sent to prison.