Is Four necessary? That might sound like a crazy question; imagine asking "Is Edward Cullen necessary in Twilight?" Or "Is Peeta necessary in The Hunger Games trilogy?" In those books, the boyfriend character is pretty important to the whole story.
But here, Tris doesn't rely on her boyfriend (and, uh, teacher, yech) when trouble comes up. She's got it covered all on her own. So, yes, Four does come in handy when Tris wants to kiss. But for most of the story, Four doesn't play a big role in Tris's problems or the plot, even. Instead, Four is mostly useful to Tris as a role model.
The bare bones of Tris's story are (a) she joins Dauntless, (b) she goes through Dauntless initiation, and (c) she foils the Erudite scheme to destroy Abnegation. Does she need Four for any of that?
The short answer is a big fat N-O.
(A) Tris joins Dauntless. Because of Four? No, she doesn't even know Four—even though Four has a very similar story to her: he was born to one of the leaders of Abnegation and then he joined Dauntless. On a zero-to-four boyfriend scale, we give him zero boyfriends for this.
(B) Tris competes to stay in Dauntless and succeeds. Because of Four? Not entirely. Sure, Four does teach her how to fight better (11.87). But Tris mostly seems to do well in the rankings because she's Divergent—that's where her true power lies. But then again, Four is very helpful here since he keeps her Divergent secret to himself when he's giving her the fear test (20.35). We'll give him two boyfriends for keeping her secret and giving her some pointers about fighting—but mostly, Tris achieves her awesomeness all on her own.
(C) Tris ruins "Jeanine's" evil scheme to use Dauntless to destroy Abnegation. Because of Four? Not so much. Tris breaks free of Jeanine and breaks into Dauntless HQ without Four helping her. In fact, he's doing pretty much the opposite. But then Four does snap out of his mind-control (thanks to Tris) and shuts down the computer. We'll give him two boyfriends out of four for that.
So on a zero-to-12 boyfriends scale, Four only has, well, four. (Honestly, we did not plan that.) Let's face it: if you take him out of the story, a lot of the big plot points remain the same: Tris joins Dauntless, passes the initiation, and stops Jeanine. Which, frankly, gives us one more reason to dig Tris: she gets it done, and she doesn't need a big strong man to rescue her.
So why is Four here? And don't say just for kissing. What happens to this story if we take Four out completely?
Well, if we take Four of the story, there is a lot less kissing, but there's also a lot less talking about how stupid the factions are. That is, Tris talks with her friends about what it's like to live in other factions (short version: no pets in Erudite); but only Four points out how narrow and dumb the factions are:
"I think we've made a mistake," he says softly. "We've all started to put down the virtues of the other factions in the process of bolstering our own. I don't want to do that. I want to be brave, and selfless, and smart, and kind, and honest." He clears his throat. "I continually struggle with kindness." (31.75)
Yeah, dude, we all do struggle to be kind and good, even (or especially) Tris. But his larger point here seems pretty smart, too: maybe it's possible to be all these good things at the same time. He's already pointed out that Tris is bravest when she's acting selflessly (24.76), so maybe these virtues can coexist or even support each other. In other words, Four is just as Divergent as Tris; and he points out to her that being Divergent might be better than being narrowly defined by one faction's virtue.
Four also provides a model for Tris as someone who is afraid, but is able to git 'er dun despite his fears. He's not without fear—people totally without fear sometimes do dumb things. (Check out the Darwin Awards if you need an example or two or hundreds.) But even though Four is afraid, he's able to do what's necessary, like climbing the Ferris wheel despite his acrophobia (12.124).
So Four is useful to Tris not because he fights her battles for her, but because he presents a model for her of how to fight and grow into the person she wants to be.
P.S. Does Four seem straight up weird to anyone else? Like, Tris notes that she's not afraid he's going to hurt her—Four isn't as crazy-evil as Eric (8.69). But when he tells her that "My first instinct is to push you until you break, just to see how hard I have to press" (24.92), we're a little creeped out. No, Four is not an abusive boyfriend, but it seems like he's got some strange impulses. Where might that come from? And how do you think it might come into play in the sequels?