Back for his third adventure in Narnia, Edmund is no longer the traitorous, sniveling little jerk he was at the beginning of The Lion, the Witch, and the Wardrobe. Instead, he has become a mature, honorable king who exhibits all the virtues expected of Narnian royalty. In fact, C.S. Lewis did such a good job transforming Edmund's character in the first book that by this point he's almost boring. The narrator of The Voyage of the Dawn Treader seems to think so, too, because Edmund never really takes center stage during any of the episodes of the adventure. Most of the time he seems to just be along for the ride, enjoying Narnia but not really contributing anything significantly different from what Lucy or Caspian can give to the voyage.
Edmund's moment to shine in this book is as Eustace's confidante in Chapter 7, when Edmund listens as Eustace describes his meeting with Aslan and his transformation from a dragon back into a boy. Like Eustace, Edmund didn't really understand good and evil when he first came to Narnia, and like Eustace, he held everyone else back. Although he doesn't say much while Eustace tells his story, it's important for us as readers to see a cycle of character transformation. Edmund has already been "converted" to the Narnian perspective, and now he can be someone Eustace looks up to and connects with during his own "conversion." (If that sounds a little bit religious, well, it is: remember, these books are full of Christian symbolism.) As Edmund himself points out, he was actually worse than Eustace when he first came to Narnia: "You were only an ass," he tells his cousin, "but I was a traitor" (7.54). Edmund's grievous transgression against his siblings and Aslan puts Eustace's cowardice and self-righteousness into perspective.
Edmund's presence also puts Caspian's own power as King of Narnia into perspective. As one of the four ancient rulers of Narnia, Edmund outranks Caspian, which allows him to speak openly to Caspian when nobody else can. Sometimes the two kings clash, such as during the adventure of Goldwater Island, but usually their friendship is strong and unchallenged.
At the end of The Voyage of the Dawn Treader, Aslan tells Edmund (and his sister Lucy) that they will never come back to Narnia. They're getting too old, and it's time for them to get to know their own world and the version of Aslan who resides there (i.e., Jesus or God). We're not surprised that Edmund doesn't need to go back to Narnia; in fact, he didn't really seem to need this particular trip. His role in the novel is to provide a foil for Eustace and Caspian; his own development was already complete.