Third-Person (Omniscient) and First Person (Peripheral Narrator)
The narration of The Voyage of the Dawn Treader, like that of The Lion, the Witch, and the Wardrobe, is a curious mixture of a third-person omniscient feel with a first-person narrator occasionally butting in. In general, the story is told in a manner we usually associate with a third-person omniscient narrator. The episodic plot tends to move between different protagonists, and the narrator dives into each of their thoughts without any trouble. When Eustace turns into a dragon, the narrator is able to tell us that "his first feeling was one of relief" (6.34). When Lucy sees Aslan's face in the magician's book, the narrator says that "she felt quite sure there had been no picture [there] before" (10.22). And when Reepicheep jumps overboard to confront the Sea People, the narrator enters Drinian's thoughts: "he liked him very much and was therefore frightened about him" (15.28). So we can tell that this narrator is omniscient and sees into every character's thoughts, describing them in the third person.
However, the narrator also sometimes uses first-person pronouns (I, me) to describe his own feelings, sometimes implying that the has met and had conversations with the characters after they came back from their adventures. For example, when Lucy is reading the magic book, the narrator says:
Some people may disagree with Lucy about this, but I think she was quite right. She said she wouldn't have minded if she could have shut the door, but that it was unpleasant to have to stand in a place like that with an open doorway right behind your back. I should have felt just the same. (10.14)
This passage suggests that the narrator knows how Lucy is feeling because she told him, which seems to contrast or even conflict with the moments when the narrator is able to see into her thoughts omnisciently. It's tempting to identify this first-person narrator with the author, C.S. Lewis, but we prefer to resist that temptation and keep biography and narrative strategy separate.
There are also some stranger moments of narration in the book using the first person plural (we, us) and the second person (you). The narrator sometimes addresses the reader directly, such as when he comments "Most of us, I suppose, have a secret country but for us it is only an imaginary country" (1.5). And occasionally these comments to the reader take on the rare-in-novels second person, such as when the narrator describes the dragon and says "Edmund or Lucy or you would have recognised it at once" (6.6).
Why such a mixture of different kinds of narration? Well, after all, the Narnia books are a mixture of different kinds of mythology – fantasy creatures rub shoulders with Christian symbols and references to Celtic and Greek legends. So maybe it makes sense that, just as the world of Narnia brings together a variety of supernatural images, the narrator of this world combines a variety of perspectives.