In the poem "The Tyger," the narrator sees a tiger in the forest at night and wonders about who might have created such a fearsome and powerful animal. In particular, the narrator wonders if the same person also made less threatening creatures, like the lamb.
In Part Three of Fahrenheit 451, this allusion to "The Tyger" acts as a metaphor to explain Montag's development. By this stage of the novel, for example, society perceives Montag with the same negative attributes as Blake's tyger: he is a dangerous criminal who has killed his captain and he is a threat to the social order because he reads books. But, just like the tyger, there is more to Montag than meets the eye. When he flees the city and meets with Granger, for instance, Montag represents hope and optimism for the future. In this way, he burns brightly as a symbol of the destruction of censorship.
Finally, Bradbury also included this allusion to foreshadow the events of this part of the novel in Montag's city. The novel closes, for example, with the bombing of the city and the total destruction of its residents. This image of the city burning brightly, then, is hinted at through this title.