Third Person (Limited Omniscient)
Our narrator in For Whom the Bell Tolls is like a little beastie which can dwell in anybody's head, but only one person at a time. The vast majority (and we mean vast majority) of the time, our perspective is that of the central character, Robert Jordan. We get to know him much better than we do any of the others, most of whom we learn about when they tell their stories to him. This frame of reference means that most of what happens in the book is seen through the eyes of an outsider who's trying to be on the inside: an American enchanted with Spain who's fighting for the Spanish Republicans, and an operative trying to integrate himself into a group of people he's just met, but must build trust with quickly.
As we've said before, the book is in part offered by Hemingway as a "Hemingway on Spain" kind of thing, and, more broadly, as "an American on the Spanish Civil War," and for this the frame is highly successful. Jordan's position in the novel mirrors Hemingway's own in reality.
Why not strictly first person, then? Perhaps Hemingway didn't want to be completely identified with his protagonist. Perhaps the third-person also makes more sense as a compliment to Robert Jordan's own personal narrative of "awakening." Since his own understanding of himself changes so much, having a bit of a distance from him allows us better to perceive the changes. A cynic might say that, if we were in the first person, the degree to which Robert Jordan dramatically changes in a few days would just come off as more bogus; plus, in the opening, when he's still kind of cardboard, he'd just be unbelievable, and rather boring.
Finally, the narration is "limited omniscient" – at some intervals, we get to go inside other character's heads too: Anselmo, Pilar, Maria, Andrés, El Sordo, Karkov, even Comrade Marty's (an interesting place). That gives us a little more perspective, and variety, than we would have if we were stuck in Roberto's cranium all the time.