The narrator seems to be a fairly empty character. We don't know the guy's name or where he comes from (although we do know he's not a native of Paris). In fact – is the narrator really a person at all?
Think about it: he appears to spend all of his time with Dupin, and his sole task in this fictional world seems to be to catalogue everything Dupin says and does, and to express amazement at Dupin's genius. He's like Dupin's personal cheering section, like "Brave" Sir Robin's annoying minstrel guy in Monty Python and the Holy Grail, but without the attitude.
In effect, the narrator and Dupin seem like two halves of the same person, with Dupin doing the talking and the narrator, the listening and recording. Consider their relationship: the two men meet because they both want the same book; they share a house in Paris; they spend all day and all night in one another's company; they enjoy playing the same game of observation (which Dupin is good at and the narrator isn't); they fill in each other's gaps.
So why not just tell the story through Dupin? Why not have Dupin as the narrator? By making Dupin the brains of the organization and separating out the narrator to tell Dupin's story, Poe steps up the suspense. After all, it's way more gripping to be told, over and over again, that a mystery can't be solved, that it's utterly baffling, that Dupin's genius is beyond compare, than it would be if it was just Dupin solving the problem all by himself.
We also need time to see the details of these murders – the shrill voice, the disordered clothes in the bureau, Madame L'Espanaye's head falling off – and to admit that we don't understand what's happening. Without this narrative time, Dupin's revelation at the end would lose all of its pizzazz. So that's why we feel that the narrator isn't meant to be a stand-alone character. After all, we get no introspection into his feelings beyond surface amazement and confusion. Instead, he's the one who puts together the database of clues that Dupin uses to solve the case.