Is the unnamed narrator from "Sonny’s Blues" a dynamic character?

In short, yes, he is dynamic—though perhaps in a more subtle way than other characters in the Baldwin canon. The essential fact is that, at the chronological start of the story, the narrator and Sonny are living in different mental worlds. When Sonny announces his intention to become a musician, his brother first asks if he wants to play classical music, when it's actually jazz Sonny is interested in. Even the type of jazz Sonny wants to play, with Charlie Parker as his model, eludes his brother. It's not that his brother is unsympathetic, as he tells Sonny he'll go out and buy Parker's records. But Sonny's interests and concerns are largely outside his frame of reference.
As the story progresses, this is no longer the case. When a reunion of the brothers finally takes place and the narrator understands Sonny's talent and devotion to his craft, something fundamental has changed. He has finally entered Sonny's world—admittedly as an outsider still, but in such a way that the quiet tone in which he ends the narrative suggests his regret that he and Sonny could not have bonded much earlier.

The definition of a dynamic character is one who changes during the course of the story. This can occur because of events that impact the character or because the character comes to a new understanding about something.
The narrator in “Sonny’s Blues” by James Baldwin is certainly a dynamic character in the story.
At the beginning, the narrator views his Sonny’s struggles with addiction as Sonny’s fault alone. Sonny and the narrator grew up in the same Harlem neighborhood and faced the same obstacles. The narrator perceived his success and his brother’s failure as the result of individual choices each of them made.
At the end of the story, however, the narrator realizes he was foolish to judge Sonny for all those years during which they had no contact. As he watches Sonny perform at the end of the story, the narrator understands just how difficult his brother’s life has been. Therefore, the narrator’s dynamic change comes in the form of an epiphany rather than a series of events.