"Manuscripts don't burn" is what Woland answers the Master, in chapter 24, when he says he has burned the novel about Pilate. The phrase asserts that a true work survives even the attempt to destroy it, and does not depend on the world that burned it.
In chapter 13, the Master tells Ivan that, attacked by the critics and sick with fear, he burned the manuscript in the stove, and remembers that "written paper burns against its will." His beloved arrived in time to save from the fire only a bundle of sheets. After Satan's ball, Woland asks to see the novel; the Master says he burned it, and Woland replies: "Pardon me, I cannot believe it," and then says the phrase. Behemoth leaps from his chair, and everyone sees that he was sitting on a pile of manuscripts.
Within the story the phrase is literal: the novel returns intact. But it says more than that. The Master lost everything because of the book, even his name, while MASSOLIT, the writers' association, has membership cards and a restaurant and no works to show. Woland does not promise that art will win in the world that persecuted it; he promises that it does not depend on that world. And the book confirms it: the Master's fate is decided on the basis of a reading, when Matthew Levi tells Woland, "He read the Master's work."
In chapter 30, the Master lets the papers burn again in the basement, without taking the novel, because he says he knows it by heart.