The narrator uses a third person omniscient voice to tell us the story. "Omniscient" means that the narrator is all-knowing, and the narrator does give us insight into all of the characters' thoughts and motives. However, the omniscience isn't always equal-opportunity – we definitely get insight into some characters more than others. In particular, we have access to Fanny's thoughts more than any other character's. But because the narrator can and very often does give us access into every person's thoughts, we can't really classify this book as a "Limited Omniscient" narrative.
We can, however, qualify this omniscient narrator as "selective." All right, let's step back and explain that. By "selective" we mean that the narrator never gives us non-stop and/or equal access to every character, even though the narrator is capable of doing so. How do we know the narrator is capable of it? Well, because nearly every character in the book, even more minor ones like Mr. Yates, get some time dedicated to their inner thoughts and feelings, however brief. The fact that the narrator often opts to not give us access to a character's inner thoughts is pretty significant, then. For instance, Mary is an especially mysterious character at times. In the scene in which Fanny and Mary rehearse a portion of Lovers' Vows together, there's a section where we hear all Mary's dialogue, and none of Fanny's, and all of Fanny's thoughts and feelings, and almost none of Mary's (18.13-24).
Our narrator's omniscience is also complicated by the fact that the narrator often seems to be speaking through Fanny, or with Fanny's thoughts and influence in mind. Some of information we get on certain characters often seems more like an explanation acting like omniscient insight. Check out the very last chapter's character round-up. Do all these character insights seem equally omniscient and insightful to you? If so, why do they, and if not, how do they differ?