Almost all the lines in PvP. Especially Marquis and ISIC. Gosh, I can’t imagine playing another PvP shooter. It would feel so dull!
The banter between Phoebe and ISIC during The Algorithm when in the frozen room before the Galactic Emperor. It’s usually an exchange between ISIC, Kleese and Ghalt, but when you play Phoebe they have their own exchange. I love the unique dialogue in this game.
Again on The Algorithm, after you defeat ISIC he asks you not to kill him because he’s “really good at stuff”
When playing ISIC in PvE, after killing a boss he occasionally says “I was a better boss fight than you”
ISIC’s quotes after killing an enemy.
When Deande sees a foreign Battleborn and says "What on Earth is that!?"
Earth? Aren’t you a Jennerit, from Tempest?
When Deande kills a thrall, “Lay down your arms lowborn”
When Mellka sees enemy Ghalt, “Kill Ghalt. But do it nicely, he’s my friend”
When Thorn kills enemy Montana, "Tiny head, big disappointment"
When she sees him, "I see something stupid ahead. Oh, it’s Montana"
When he sees her, “WATCH OUT, SHE’S ANGRY!”
My favorite little touch is the callouts characters have with each other. Some of my personal favorites being:
Montana to Rath:
“Guys, I just killed a freakin’ vampire!”
“It’s one of those Live Action Role Players!”
“Didn’t think this was a costume party!”
Orendi to Anybody:
“REYNAAAAAAA”
“Guess who’s gonna regret saving my life? Spoilers, it’s Phoebe!”
“The fancy robot is mine!”
Marquis to Peasants:
“Lower class individuals, inbound!”
"Hobo shpotted!"
Hoodini coos “Jah, Jah, I see zem!”
All the character dialogue is gold, I hear a new character quote every few matches and almost all of them get a chuckle out of me.
Honorable mention goes to backing quotes.
“Fleeing like an Olympic track baby!”
“We are not retreating! We are…yes, we are retreating.”
“Back to the only home I haven’t turned into digital hellfire!”
Solid. Gold.
I think my favorite little thing is ISIC’s quick melee. Sometimes instead of the usual take that or grunt, he’ll just say “Fellah!”
I just know someone, somewhere, possibly the voice actor, thought “Gosh, they sure do say Fellah a lot in his game.”
Was unlucky to get killed by a sentry on incursion. It then proceeded to taunt me by flipping me off jack-in-a-box style.
But yeah, the escort missions have by far some of the funniest dialogue with wolf and sentry. Sentry is personal fav, with lines something like:
“My cousin Geoff gets to be a spider king, while I’m being kidnapped”
When you deliver data bots: “365,568 cookbooks. Suuuure, that will be useful”
Not really story related, or anything…
But I love it when about to start a match with random people, and we are all picking our characters and skins…
And then everyone changes to have the same skin on (eg, the blue one)…
None of us have mics (or are using them at least), so we all hit our taunts, as if to high five each other…
I don’t even care if we end up losing, because, got that team spirit and we don’t even know each other, haha.
Another thing I love is the voice dialogue in the lore for Toby and Orendi.
Ambra’s voicelines. Especially her laughters. Also her quotes that includes “my dear”. She pronounces those two words in such a way that… never mind. Actually, my very favourite little thing is Oscar Mikes’ purple tutu that I’ve seen in the games’ trailer.
Ghalt seeing enemy Ghalt: "Kill me! N-No, I mean the other me!"
OM seeing enemy OM: "Evil twin spotted!"
Rath killing Miko: “What the hell did I just kill?”
I like how we don’t have to click before the title screen starts to load your account. It was annoying in BL2 and BLTPS.
I also like that skills have very short cooldown time when you’re with Nova before the match starts.
Phoebe’s “Run, run from the little girl hahaha!”
Incursion and the sentry giving you the crank up middle finger.
Okay so a bit of an odd one,
On one of the missions, i think its sabotage, just before you enter the large factory complex, if you look up there is a smoke stack with smoke pouring out. I cannot get enough of this smoke stack, i just stare at it in a trance, my team are dying, i am the healer, and i am staring at some animated cartoon smoke 
When I read ISIC’s lore, he joins pro wrestling. Does an interview before his fight with El Dragon. When asked what ISIC’s strategy is he replies, “Oh, I’m going tear his f***king arms off!”
…and that’s why El Dragon has robotic arms.
I’M STILL LAUGHING ABOUT THIS!
Ghalt singing when he throws his chain. Mainly the line
“Step into my office”
Boldur “Im the only one the Forrest talks to” in a deep voice “And the Forrest says -Hallo” with a happy tone 
I like how Geoff and chronicle are cousins
I like the mortal kombat reference when he yells: “get over here” 
I’ve heard one where in PvE the sentry was hitting on thorn. Hahaha so funny
That akward moment when boldur talks about melka’s hair…
I really wish we could have a good list of all the moments in PVE where we get contextual dialogue when certain characters are about. The most obvious bit is Wolf in the Void’s Edge where he asks the people what they will be doing after the mission is over, but it’s less obvious in other missions. In the Saboteur, there was a couple moments where Toby was asking Nova where to go to next, and being very polite, keeps saying “Where to Miss Nova?”
I think when the door is broken down in The Archive is another place where there seems to always be a character line, but again, it gets murky when characters like Ghast, Kleese, Deande and Mellka are on the team and they also serve as the mission NPCs too.
Lol isic has the best line