I would say the bonfire is a fairly simple bit of symbolism from the novel. The bonfire travels with the circus and remains lit at all times. It is lit during performances. It is lit when the circus is on break, and it remains lit while the circus travels from location to location. Perhaps it is the circus's "life force"? I like to think of it as similar to the heart of a human being. As long as a person's heart continues to beat, blood and all of the nutrients that it carries will continue to be delivered to the intended locations. That person will live (yes, I know there are other ways to die with a healthy heart). In the same way, the night circus will continue to thrive as long as that fire keeps burning.
"The bonfire never goes out. The flames never falter.
Even when the circus moves it is not extinguished, moved intact from location to location. Smoldering the entire length of each train journey, safely contained in its iron cauldron.
It has burned steadily since the ceremonious lighting on opening night."