Third Person (Omniscient); First Person (Peripheral Narrator)
If we look only at the text of "Teddy," we have an uninvolved third-person narrator who tells us what's going on and what's being said – not without bias and interpretation, but excluding any forays into the characters' heads. We stick to Teddy for most of the story, but the camera freely switches to Nicholson for the final scene.
If we go outside of Teddy, into some of Salinger's other stories, we find some interesting info. Buddy Glass, one of Salinger's fictional characters and first-person narrators of several other Salinger stories, actually claims to be the author of "Teddy." This is an interesting addition, because it means we can use "Teddy" to understand more about the character of "Buddy."
To do so, however, we have to read some more Salinger, or at least check out "A Perfect Day for Bananafish" and Franny and Zooey on Shmoop. You can also read Teddy's "Character Analysis," in which we discuss his connection to Salinger's fictional Glass family characters, including Buddy and "Bananafish" protagonist Seymour Glass.