Who is the narrator, can she or he read minds, and, more importantly, can we trust her or him?
Well, none—it's a play.
Major Barbara is a play, so there isn't some overarching narrative voice that frames or makes sense of the dialogue for us. That said, Shaw's stage directions do give some character background and do more editorializing than most, as we've already kind of alluded to in "Characterization." Also, given the extreme amount of airtime that Undershaft's ideas get (and the fact that he's clearly the hero of the play), you could kind of argue that his ideas, perspective, and "slant" really end up being the play's as well. But technically, there's no narrator, so no official point-of-view.