Burne is the brother of the confident and charismatic Kerry Holiday. When Amory first meets Burne, Burne seems like so much of a go-getter that he has no time for anything other than doing well in school. As the narrator tells us, Burne's dream is to become the editor of Princeton's premier paper:
Burne, fair-haired, silent, and intent, appeared in the house only as a busy apparition, gliding in quietly at night and off again in the early morning to get up his work in the library—he was out for the Princetonian, competing furiously against forty others for the coveted first place. (1.2.80)
Burne knows how valuable this position at the Princetonian would be for his social status, and he nearly tires himself to death trying to achieve it. Unlike his brother Kerry, Burne doesn't think life is easy, breezy, and beautiful. Instead, he is:
… a man who gave an immediate impression of bigness and security—stubborn, that was evident, but his stubbornness wore no stolidity, and when he had talked for five minutes Amory knew that this keen enthusiasm had in it no quality of dilettantism. (1.4.28)
In other words, Burne means business, no matter what business he's talking about.
Ultimately, Burne takes his desire to rise to the top of Princeton society and turns it into a revolutionary desire to tear the whole thing down. He wants all of Princeton's exclusive clubs to be done away with, and he refuses to fight in World War I on the grounds that it is a stupid war, and morally messed-up.
As Amory sees at one point, "Amory saw Burne and knew at once that argument would be futile—Burne had come out as a pacifist" (1.4.289). So that's that. Burne eventually disappears. But Amory continues to envy him as a guy who can stand up for his convictions instead of taking the easy way out—which is Amory's favorite way out.