Given the definitions of static and dynamic, I consider the narrator to be a static character in this short story. First, think about how insignificant the narrator is to the story. He is never even named. Truly, his only purpose is to tell the story. Given the fact that the events have already fully happened at the time he is telling the entire story, it should be assumed that though he attempts to be as emotionless and as objective as possible, he begins the story fully knowing the tragic outcome at the end. He is already affected by his experience at the Usher house, which means his demeanor and tone, no matter how much he may attempt not to let them, are affected by this experience from the very start. They do not change as a result of events because the story is told after the fact.
The narrator begins the story as a character who is diplomatic (in visiting a childhood friend) and polite (given the strangeness of such a friend). He tries not to be judgemental and though he is descriptive in his story telling, he does not interject strong opinions of the strange events, except to emphasize that they really happening and that he was alarmed. Even by the very end of the story (through a consistent increase of bizarre events and fear) the narrator maintains the same sense of calm in his voice as he had at the beginning.
To be honest, this story is just short enough that I would argue all of the characters in it are static. The Ushers end tragically because they began tragically. They are strange and pitiable from beginning to end. The narrator wishes to be viewed as someone who is as disconnected from this family as possible. He remains polite but in this endeavor, he succeeds.