Black Mage and Fighter of 8-Bit Theater. Fighter thinks BM just likes to kid around, even when BM stabs him in the head. (Fighter is dumb and gullible and more or less indestructible, so Black Mage's stabbing attempts either deal no damage at all anyway, or on the rare occasions they do, the resulting brain damage once even made Fighter smarter! Either way, he never seems to take offense.)
Fighter: Black Mage and I are bestest buddies!
Also, by the time of the epilogue, they're still together as adventuring partners despite the fact that the rest of the team has since broken up. Not like Black Mage is happy about this...
Enigma and all seven of his brothers in Dragon Tails. Despite threatening death to any who touch his lake and using memories of his brothers' past idiocy to fuel his rage-filled fighting style, Enigma will do pretty much anything to protect his family, risking his own life and limb to ensure their safety. There's also Lemuel and Norman. Norman often schemes to use Lemuel as a disposable pawn in his quest to conquer the world, or as a bodyguard or slave. Lemuel...has a pet food dish labelled "Normie".
Faye and Dora in this strip. Also many others, but in this one Raven describes it: "Sometimes I can't tell if you guys are friends or if you hate each other."
At Marten's dad's wedding, his mother meets an old friend and they trade really nasty barbs. Claire looks a bit shocked, but Marten just comments that his own taste in friends (ie Faye and Dora) suddenly makes sense.
Pretty much everyone in Something*Positive insults everyone else constantly and to their faces, as Davan explains here. Sometimes it's hard to see the difference between the way they treat their friends and the way they treat people they'd be happier without, even for the characters. There's more than one genuine hatred mistaken for affection relationship. Kharisma and Mike especially are prone to being on the unliked end of them. Then again, all the core cast are in vitriolic relationships, so it's hard to blame them for the mistake.
PeeJee: Sometimes I hate you so much I worry it might be love. Davan: Talk like that is why people speculate about us, y'know.
Worth pointing out as well, though PeeJee, Davan and Aubrey collectively torment, insult and play with each others emotions or lack there of to a degree, all three would literally rip through hell to get at someone who'd wronged the other considering even Davan forgives Eva before the noose is loosened around her neck by any of his friends short of Jason.
Largo and Piro from MegaTokyo are total opposites in many ways, with Piro spending a lot of time making snarky comments about his friend's loose sanity and total lack of social skills. For his part Largo seems to think of Piro as a weak hearted, easily manipulated fool who is easy prey for the forces of evil (They're both right). This makes the moments of true friendship they show all the sweeter.
Roy Greenhilt and Elan from The Order of the Stick. Elan seems to be cheerfully oblivious to... well, a LOT of things, actually, but especially to how annoying Roy finds him.
Vaarsuvius and Belkar are a pretty good example of the less benign version. They constantly attack, insult, sic animals on, and berate each other, but both have saved the other at least once, and have risked themselves for the other. And then there was the event....
Tony and all his friends in Charby the Vampirate but especially Zeno. While Zeno considers Tony his best friend, this hasn't stopped Tony from delighting in Zeno's rejection and humiliation. Menu is a major offender here too.
Similar to the Black Mage/Fighter example is that of Dominic and Steve in the Magic: The Gathering webcomic UG Madness. As the author himself says, the best thing about Steve is that "he is far too intent on becoming your friend to ever get angry with you. You can even smash his car with a hatchet- it's cool."
Debatable since many of the characters actually hate each other and have tried to kill each other almost as often as they've aided each other, and when one of them is in trouble, it often takes some convincing before another one will help, and he or she might not be too pleased about helping.
Penny Arcade has Gabe and Tycho, who constantly share petty insults and at times have been shown to violently murder each other.
Writing The Rant attached to the comic, Both Mike Krahulik (Gabe) and Jerry Holkins (Tycho) have depicted themselves as a parasite riding on the talent of the other.
In Girl Genius Gilgamesh and Tarvek fluctuate somewhere between "best friends" and "hate each other's guts". Agatha ironically predicted they'll like each other (due to being Not So Different) before discovering they're already acquainted.
Gil: (with Tarvek in a choking hold) Listen to everything this duplicitous snake says!
Casey and Andy room together, and are best friends since high school. They also kill each other on a regular basis. (not try to kill but kill). On the other hand, they have saved each other from deadly threats.
Our Little Adventure: Rocky and Angelika. They snark and bait eachother on a regular basis, but they are obviously very good friends, and have protected eachother since before meeting up with Julie and Lenny.
Dorothy and Elanor in Kevin & Kell tend to be antagonistic to each other (once even indirectly causing a broken hip), due to natural prejudices (Dorothy's a rabbit, Elanor's a wolf), disagreements over how to manage Aby's business, and their mutual granddaughter's diet (Dorothy is trying to push a more vegetarian diet on the naturally omnivorous Coney), but they agree that they are each other's best friends, partially because between old age and predation, they don't really have anyone else left to call friends.
The Trolls in Homestuck are largely a species of Jerkasses, and their form of friendship is largely being jerks to each other, but they more or less get along. They ultimately grow out of it and into genuine friendship...offscreen.
Even among the trolls, Karkat is a standout example. He cares deeply for others, but he's also a perpetually angry jerk, meaning that this kind of friendship is about the only kind he has. Lampshaded by John when thinking about how he'll greet his friends after a long journey:
John: ...and I'll push Karkat down a flight of friendship stairs!
The most textbook example for Homestuck has to be Karkat and Sollux:
Karkat: I GUESS THESE CONVERSATIONS WE HAVE DO GET KIND OF EMBARRASSING IN RETROSPECT.
Karkat: ARE WE NOT FRIENDS ANYMORE BECAUSE OF STUFF I SAID.
Sollux: eheheheh you LIITERALLY a2k me that every tiime are you jokiing.
Dave and Karkat are another particularly good example. They're apparently "bros" now, and Karkat even refers to Dave as his "best friend" at one point, despite the fact that they spend virtually all of their shared screen time insulting each other. And they might possibly be something more...
Rose's relationships with everyone tend to be based on a mixture of friendly sarcasm, biting sarcasm, semi-gentle mockery, and armchair psychology.
Out There is something of a World of Snark, so many of the close friendships therein follow this (Miriam and Sherry, Sherry and Steven, Araceli and Rod, James and Chuck), or think they do (Miriam and James, Sherry and Clayton, Miriam and Clayton). The schizophrenic nature of these relationships is well represented here; Miriam and Sherry's volatile friendship borders on a parody of this trope at times; when they're not sniping at each other, they're often caressing each other.
Anthem and Gilgam in The Water Phoenix King — they're really the only two people who can, or would dare, to tease each other, until recently. Seriously, he's the Really 700 Years Old, battle-scarred Proud Scholar Race Guy who helped defeat her people's side in the war, and she's the Ax-CrazyDark Magical Girl who goes Dark Willow at the drop of a hat, and they both know all this about each other almost from the start and don't care. A more perversely-beautiful friendship you could hardly imagine.
Ambassador Whree: I... have never found a sentient I enjoy hating more than I hate you.
This works on the scale of species: Klegdixals seems to be easily irritated (it takes a lot to make one say "I... I feel positively un-unhappy!") and have jerkass reputation in turn. Humans happily drive them mad or blackmail, given a chance.
I predict that a Klegdixal/Humanity partnership will produce a unique dynamic, one that to outsiders will appear quite doomed, but...
VG Cats: Aeris and Leo, the main characters, do stuff to annoy the hell out of each other. It even got to the point where, in one strip, Aeris went back in time to abort an unborn Leo out of existence. But all said and done, there are moments when they show they actually do care for each other—"The Perfect Gift" being the main example.
In Nodwick, this dynamic is clear between Yeagar and Arthax, who consider each other equals but take every opportunity to snipe at the other for being (respectively) Dumb Muscle and a Mad Scientist. Arthax and Nodwick exhibit this dynamic as well, as Arthax seems to respect Nodwick's common sense even though he constantly abuses him. Yeagar and Nodwick are closer to mutual Sitcom Arch-Nemesis status: Yeagar is usually the first to go for the "mistreat the henchman" solution, while Nodwick frequently snarks at, belittles and annoys Yeagar with verbal barbs. Although they probably hang out more than any other combination of the group, Comedic Sociopathy usually rules the day between them.
Tower of God has a plethora of examples. First of all, Koon Aguero Agnis and Rak Wraithraiser are a blue-haired noble and a giant humanoid Alligator respectively, and they do not get along from the very start, yet still maintain a bond of trust and rigid understanding after their team is dissolved. Ronin Hatsu and Lizard Girl Anak are also pretty close despite fighting a lot. Then there is Evan/Yuri. He gets thrown off cliffs a lot. And finally, there are Androssi and Anak who play this out like an Ur-example.
Sara and Jane of Nobody Scores! (dis)function like this. They're constantly snarking at or pranking each other but they've been friends since high school and are now roommates. The horrors inflicted on Beans are more an afterthought.
Nebula: Mars finds Earth to be intrusive, annoying, and far too invested in everyone else's business, and frequently tells her to go away; she just wants to have friends and keeps at it anyway, refusing to take it personally. Mars does care about her though, and when he thinks things are getting outright dangerous he goes to her first to try and warn her before anyone else.
In Cobweb And Stripes, this is the relationship between Betelgeuse and Jacques. It doesn't seem, the way they interact in their earliest scenes together, like they particularly like one another. But as in the cartoon, Jacques and his wife Ginger are Betelgeuse's only real friends in the Neitherworld and he's actually rather (quietly) fond of them.