Opinion is somewhat mixed about this, although the belief that she will survive her encounter with "Lady Stoneheart" seems more common. The key matter is just what the word she shouts is. There are those who think she merely shouts the name Jaime out a last time before dying, but another word presents itself in the text: "sword" This is one of the two choices that Brienne is given, the noose (death) or the sword (promising to kill Jaime Lannister). Why would Brienne do this, despite her feelings towards Jaime? Just before she shouts, she looks to Ser Hyle and Podrick as they’re hanging and dying, and it’s only after this that she shouts. Those who believe that the word is "sword", or some variation on it, take this to be the reason as to why she does it: if it were her life alone, she might have accepted death, but she cannot let her companions die. The final descriptive sentence before the shout, "nothing had ever hurt so much," may well refer to the mental anguish at choosing the lives of herself and her companions over that of Jaime Lannister.
The Jaime chapter in A Dance with Dragons seems to settle matters, with Brienne appearing to lead Jaime away from his companions and disappear with him. However, it’s been proposed that Brienne did indeed hang, only to have Thoros or Lady Stoneheart somehow resurrect her. While we do not believe this is the case, that cannot be definitively confirmed or denied with the evidence at hand.