FDR would have a field day with this book. As you probably know, FDR (a.k.a. famous American President Franklin Delano Roosevelt) was famous saying "there is nothing to fear but fear itself." Lupin gets his FDR on in this novel when he tells Harry that it's "wise" (8.4.53) to be most afraid of fear itself. Aside from letting us know that FDR would totally be a Harry Potter fan, that scene also highlights the lack of Voldemort in the novel. (Sure, we keep bringing up Voldemort's absence, but his MIA status is a hugely important thematic detail. See, this novel is more about internal battles and character development than external fights; having lots of big external conflicts would detract some from that.)
Still, in Book 3, it's the Dementors that really highlight the theme of fear. These creepy, reaper-like characters embody fear, death, and darkness. The Dementors are giant walking (or gliding) metaphors for the world that Voldemort represents. Luckily, we also get the flip side of fear, as represented by the Patronus charm. Light is literally used to battle darkness here. It's worth noting that being around Dementors reminds people of horrors from their past and makes them forget who they are. The past itself can haunt and create fear. And forgetting happy memories can erase a person's identity, which implies that people aren't defined solely by their fears or their inner darkness. So while details like a character's boggart give us important insight into that character, the novel also cautions us against identifying people just by their fears.