EDI and Joker in Mass Effect 2 are this trope for a while, to the point where Joker fractures his thumb hitting her mute button. Over time their relationship improves, however, particularly after he breaks Cerberus' control over her. Not only does this enable her to save the ship from a Collector attack, it also makes EDI's personality more pleasant.
Garrus and Tali are this as well, if their lift conversations are anything to go by ("I'm pleased that the imminent destruction of all organic life has improved your career opportunities," for example). However, they also seem to be very respectful of each other's abilities. As of the third game you could consider this Belligerent Sexual Tension, as well, since the two of them get together despite their snarking at one another, provided neither one is already in a relationship with Shepard. And if female Shepard is in a relationship with Garrus, a drunk Tali will have a daydream about Garrus offering her a threesome.
Mordin Solus is also this with STG operative Kirrahe. Mordin speaks of Kirrahe fondly when the latter's role in the first game is mentioned, but calls Kirrahe a "bit of a cloaca". In the game's cut content, Kirrahe's email to Shepard also speaks highly of Mordin, although he also calls the latter a "bit of a cloaca". note (In some animals, a cloaca functions as both genitalia and waste disposal, so among salarians, it's a way of calling someone both a dick and an ass at the same time.)
James Vega and Lt. Cortez in Mass Effect 3. When you're speaking to one, the other will likely be heckling them from the side, typically Vega towards Cortez.
Mass Effect 3 also shows that Garrus and Wrex have developed into this. Having Garrus in your squad while talking to Wrex on Sur'Kesh is hilarious because of the way they just go at each other. With big smiles on their faces the whole timenote well, a smile on Wrex's face anyway. Turians don't have the mouthparts to smile.
Garrus and Male Shepard have this type of relationship. Best shown in Garrus' recruitment mission in the second game, where after having sustained heavy facial scarring from being struck in the cheek by a missile, Garrus jokes that he was getting tired of all the female attention, so at least now Shepard can get some attention from the ladies. One of Shepard's options for a first move, meanwhile - and this comes with either gender - is to don a broad smile and go "hell, Garrus, you were always ugly; slap some paint on it and no-one will notice".
Despite an intial animosity due to the disastrous first contact between them, the Turians and Humans eventually have evolved into this type of relationship by the third game, where they are the first to mobilise and combine their forces to fight back against the Reaper invasion. While they still continue to heckle one another, it's clear both sides have a grudging respect for their abilities as soldiers and have got each other's back.
To call them "best buds" might be stretching it a bit, but the party during the Citadel DLC paints Miranda and Jack as this. They're still constantly flinging insults at each other, but even Jack admits that they're just messing around. According to her, Miranda fights better when she's pissed, so she tries to keep her primed. Miranda later admits that she finds Jack's Character Development extremely admirable, for which Jack sincerely thanks her... only to respond that the thing(s) she admires most about Miranda are her tits. Even if you encourage their rivalry, it ends in the two of them admitting that the other is "fun to hate."
Most of the krogan conversations you hear on Tuchanka in the second game are either this or two people who genuinely can't stand each other; it's kind of hard to tell.
In Citadel, this kind of relationship is all over the place. While Jack and Miranda are the most obvious, you can get interactions like Shepard making jokes about Kaidan's cooking, or Miranda affectionately mocking you for the whole fiasco in the first half of the DLC.
The two goddesses of the Moriya Shrine, Kanako and Suwako, have this relationship. Long ago, Kanako took over Suwako's shrine after defeating her in a fight, but since Suwako was the only one who could control the local curse gods, this made the people fearful. Kanako had no choice but to have joint ownership of the shrine with Suwako to ensure the curse gods wouldn't get out of hand, though Suwako insists that it's still hers. They've lived together in the same shrine ever since, still considering each other rivals but not so much as they used to.
Strangely enough, Kaguya and Mokou could be considered this even though they try to kill each other constantly. Since they drank the Hourai Elixir, the very aspect of death has been removed from their lives, and they very well might outlive every other being in Gensokyo. Partially because of this, they are the only ones who can understand the other's plight with this Immortality. Cage in Lunatic Runagate has Mokou explaining to Keine how terrified she was when she heard about the Moon rocket in Silent Sinner in Blue. She thought Kaguya was taking it to the Moon and was going to leave forever.
Guillo and Milly in Baten Kaitos Origins are a cross between this and Fire-Forged Friends; they outright despise each other at first, and despite warming up to each other, they still hurl insults every chance they get.
In the Shin Megami Tensei Persona series, it's quite common to find a couple whose friendship is seemingly based on jabs and cheapshots.
Masao "Mark" Inaba and Kei "Nate" Nanjou from the Revelations: Persona.
Eikichi Mishina and Lisa Silverman in Persona 2: Innocent Sin. Both have derisive nicknames for each other, and argue constantly. At a certain point in the game, Lisa is revealed to be with an unsavory character, and Eikichi states that they better go find them before Lisa murders the guy. Another character (the Cool Big Sis) mentions that Eikichi is just saying that, and he really seems worried about Lisa.
Appears a lot in Persona 4 as the main cast seem to love annoying, teasing, and bickering with each other. Some notable examples are:
Yosuke Hanamura and Chie Satonaka. Some fans like to interpret their relationship as Belligerent Sexual Tension, but regardless, it's hard to deny that they're close.
Yosuke and Kanji Tatsumi, too. Through a combination of Yosuke's teenager thoughtlessness, mild homophobia and his teasing personality along with Kanji's hair trigger temper and insecurities about his sexuality causes more than a few heated moments between the two. But like the rest of the Investigation Team, they are Fire-Forged Friends through and through.
Kanji also tends to bicker with Rise Kujikawa whenever they directly address each other.
An example from the mainline SMT games, Walter and Johnathan from Shin Megami Tensei IV, being from two differant caste classes, they often but heads on how the other should act as a Samurai, but they do often help each other out and are companians. That is until they encounter Lilith and the Archangels. Their friendship is momentarily back again when they and Flynn travel through the alternate Tokyos, only to completely shatter once they come back and Walter and Johnathan meet Lucifer and Merkabah, respectively.
Duncan and Sand from Neverwinter Nights 2 are all over this trope. While the term "best buds" might be a bit of a stretch, they are what one would call friends, and the general fan consensus is that if they ever were to stop insulting one another, they would probably just... implode... from the stress.
Duncan: Sorry I pointed out the fact that you were a charlatan, Sand. Sand: And I didn't mean to bring up your excessive drinking, Duncan, and your long list of failed aspirations. It was uncalled for.
Daxter and Pecker in Jak and Daxter usually show nothing but scorn for each other, but Daxter has no problem letting Pecker drink at his bar for free. They even teamed up to try to save Jak from his banishment in Jak 3. It didn't work, but it's the thought that counts.
Gan Ning and Ling Tong in the Dynasty Warriors series. Despite being the blue oni to Gan Ning's red oni, Ling Tong is the more vitriolic of the pair - which is justified in that Gan Ning killed Ling Tong's father before the two became comrades. Eventually they settle into ragging on each other's performance on the battlefield and griping at each other whenever an Embarrassing Rescue is called for.
Jahiera and Aerie in Baldur's Gate 2. Much of their dialogue involves them arguing or otherwise exchanging barbs with each other (especially if Charname's romancing one of them), but in the end they back each other up in a fight.
Star Fox. Fox McCloud and Falco Lombardi: although Falco is definitely more so. Fox takes playful shots back on occasion.
They were even more vitriolic during the original Super NES incarnation's comic, when Falco was even more cynical, Fox was even more prone to raising his voice, and the two were known to occasionally deal with their disagreements by beating the living shit out of each other, even while simultaneously expressing how much they cared about each other.
Nick and Coach from Left 4 Dead 2 sit somewhere between this and Fire-Forged Friends. At the start, each of them sees the other guy as an idiot, but after a while, they come to have a bit of respect for each other, even though the way that Nick admits it is by saying Coach has "the stupidest idea I've ever agreed with".
From the original Left 4 Dead, Louis and Francis seem to get along like this; Francis's hatefilled world view and Louis's optimism clash, but if one of them gets killed the other is visibly upset.
Baird and the rest of the team in Gears of War, especially between him and Marcus. The two barrage each other with insults, but would risk their lives to save each other.
Sheena and Zelos in Tales of Symphonia. Though he's always teasing her and she says she can't stand him, they both care a great deal about each other.
Tales of the Abyss has a case with Jade and Peony. Particularly notable in that Jade, in his role as The Gadfly, teases everyone, but actually turns it up for Peony, and Peony is the only character in the game who ever manages to tease him back just as hard.
Tales of Vesperia has a mild case, with the Heterosexual Life-Partnerspolar opposites: Yuri and Flynn. Despite the teasing, they genuinely respect each other, and their insults are typically either endearing or something they like about the other. Of course, this is conveniently out of the other's hearing range. ... Otherwise, expect them to bicker like an old married couple, and not stop until someone else interrupts them. The prequel movie added a lot more vitriol, but there it didn't look like they were friends at all.
The movie deals with the time when they started to pursue different paths; that's why it seems they can't get along.
In Dragon Age II, there's Aveline and Isabela who constantly bicker, calling each other things like "whore" and "man-hands", but hang out with each other in their free time, and by the end of the game it's obvious they care about each other. At the end of Act 2, if you opt to hand Isabela over to the Arishok, Aveline will object to the point of getting rivalry points for the act. Their relationship condensed:
Aveline: I'm the captain of the guard. I'm loyal, strong, and I don't look too bad naked. Isabela: Exactly. And if I called you a mannish, awkward, ball-crushing do-gooder, you'd say... Aveline: Shut up, whore. Isabela: That's my girl.
Carver and Varric have a similar relationship:
Carver: Varric. Varric: Carver. Carver: Still think you're helping while burying us in debt to your brother? Varric: Still riding side saddle while bitching at your betters? Carver: Drinks later? Varric: Never miss 'em.
Varric (again) and Aveline have frequent spats about the former's "guard serial" and who was the model for it.
Rivalries in general are like this. Hawke acts extremely antagonistic to his/her Rivals, but in a way that forces said Rivals to change and grow as people. The Rivals in turn will snark back and sometimes even lash out, but full Rivals will be just as loyal to Hawke as full Friends.
In Origins, many people interpreted Morrigan and Alistair's relationship to be something like this, even sparking a strong shipping base based on their seemingly-Slap-Slap-Kiss interactions, until Darkspawn Chronicles went and jossed the fandom's interpretations, saying that they truly just plain hated each other. could've fooled us, I guess.
Saved by the fact that the Darkspawn Chronicles DLC was never meant as canon, but defined as alternate reality.
Morrigan does, however, say late in the game while proposing to sleep with Alistair to become pregnant with a Gray Warden's child that will become host to the Old God that Alistair despises her, which is why she is going to you and not him.
Sonic the Hedgehog often criticizes Knuckles' gullibility while Knuckles is always too short-tempered and reluctant to admit that Sonic is his friend. However, when things get tough, Knuckles shows that he's one of Sonic's most reliable allies and is easily his closest friend after Tails.
Sonic and Shadow are both this. This was evident in Shadow the Hedgehog where Sonic treats Shadow as a friend while Shadow is rather irritated with Sonic's friendly attitude (granted that Shadow still had amnesia and doesn't know Sonic that well at least until the end of the game). However, there are times where Shadow does come through for Sonic when he saved him from being attacked by Silver in Sonic the Hedgehog (2006), and he does have a great deal of respect for him.
Shadow and Rouge also qualify, but this is mostly because of Shadow's aloof and distant nature which Rouge gets annoyed with sometimes.
Cole Phelps in L.A. Noire is this to all of his partners.
Disgaea: Hour of Darkness gives us Laharl and Etna. He's an arrogant spoiled brat prince who demands utter loyalty and obedience from his vassal, and she is a tsundere Servile Snarker who claims she is constantly one last straw away from bumping him off and taking the throne for herself. They argue and insult each other constantly. But deep down Etna is devoted to making Laharl the best ruler he can be, and deeply respects how much he has matured, and Laharl truly admires her ambition and competence and basically gives her permission to scheme against him, because if there's one person who deserves to overthrow him it's her.
Disgaea 3: Absence of Justice has Mao and Raspberyl. One an Honor Student (meaning total jerkass) and one a Delinquent (goody two-shoes). They can't stand each other, insult each other, fight each other, and by the end of the game would clearly gladly lay down their lives for each other. They actually end up running the school together.
Heavy Rain show a deconstructed example in Norman Jayden and Carter Blake. They got themselves a good start when they first met, but as the game goes on, their continuous fighting, mostly derived from having different methods, eventually drives them apart as they go on separate ways in catching the Origami Killer.
Kyo Kusanagi and Benimaru Nikaido start as this in The King of Fighters. The fact that their friendship began few after a very young Kyo defeated Benimaru in two consecutive martial arts tournaments doesn't help. As they both grow and mature their banter is dialed down quite a bit, but they're still ready to spar whenever they can.
In the KOF:KYO media, Kyo and Kensou have some shades of this as well.
Dylas and Doug in Rune Factory 4. They constantly bicker, with Doug having no reservations about pressing Dylas' Berserk Button, and struggle to outdo one another in any sort of competition. But in the end, they still leave each other (anonymous) birthday presents every year.
Plenty occur in Fire Emblem Awakening due to the massive amounts of possible supports between characters. One example is the Female Avatar and Lon'qu. She hassles him because he refuses to take her seriously and he finds her antics frustrating when he thinks she refuses to respect his phobia of women, but by A-support they get used so used to each other, he's conditioned to be distracted from his phobia around her.
Assassin's Creed IV: Black Flag has Edward Kenway and Adewale's relationship verge on this. Adewale has no respect for Edward's endless greed while Edward ignores all of Adewale's idealistic suggestions. They eventually reconcile when they both become Assassins.
Galuf and Faris are this in Final Fantasy V. They bicker a lot and goad each other, both of them having pretty rough manners as a Cool Old Guy and Pirate respectively. Even when Galuf is dying, Faris' plea for him to live is "Get up, you old bat! Quit playing around, this isn't funny!"