At first glance these two labels might seem to be contradictory: how can the tone of a book be both playful and solemn? Aren't those opposites? Well, yeah, but The Voyage of the Dawn Treader is definitely both of these. The narrator is often playful and lighthearted, especially toward the beginning of the book, when he's setting the scene and introducing us to the characters. Many of the narrative asides have a teasing, jolly quality; for example, when describing the painting of the ship that hangs in Lucy's bedroom at her Aunt Alberta's house, the narrator butts:
By the way, if you are going to read this story at all, and if you don't know already, you had better get it into your head that the left of a ship when you are looking ahead, is port, and the right is starboard. (1.7)
Over the course of the book, as the Dawn Treader gets closer to Aslan's country, the narrator's playfulness gives way to a more and more solemn, awe-filled tone:
After that for many days, without wind in her shrouds or foam at her bows, across a waveless sea, the Dawn Treader glided smoothly east. (16.3)
We like to think of this transition from playful/teasing to awed and inspired as the process by which the narrator fades away, leaving us to gaze on Aslan's country on our own. As he gets less playful and more solemn, his personality fades a little and lets the story speak for itself.