Dinobot and Rattrap from Transformers: Beast Wars start off hating each other and argue constantly. The bickering continues after they grow close, particularly after Optimus forces them together to help save Rhinox. When he hears them still sniping each other later...
Optimus Primal: It never ends, does it?
In fact, it's when one doesn't snark back that the other knows something is wrong.
South Park: Stan, Kenny, and Kyle's relationship with Cartman. They love and hate each other because Cartman is a sociopathic, deranged racist who makes everyone do his work. As Chef puts it, Cartman is "their friend, whether they like him or not."
Lampshaded in one episode where a formerly homeschooled kid with no concept of male friendships begins hanging out with them. He mistakes the boys for bitter enemies, and they are forced to give him a lesson in "guy talk" (a.k.a. the art of harmlessly insulting someone you actually like.)
Similarly, in the episode where Token gets upset about being 'the only rich kid in town', the boys explain to him that they all rip on each other all the time. Like they rip on Kenny for being poor, or Kyle for being a Jew, or Stan for being in love with Wendy, or Cartman for being fat, or Cartman for being stupid, or Cartman for having a whore for a mom, or Cartman for being a sadistic asshole...
A one-sided case in SpongeBob SquarePants. SpongeBob is utterly convinced that Squidward is his best friend while Squidward makes it obvious he's anything but.
Except for whenever Squidward assumes SpongeBob is dead, and then he mourns him. When SpongeBob inevitably comes back, Squidward reverts to being a jerk, but we all know as annoyed by SpongeBob as he is, deep down, he likes him.
Squidward: SpongeBob, no matter what I've said, I've always sort of liked you! SpongeBob: Squidward, I used your clarinet to unclog my toilet! Squidward: Huh?
Another telling example is when Mr. Krabs sells SpongeBob into damnation with the Flying Dutchman for 62 cents, and Squidward is genuinely shocked and angry.
Also, when Squidward moves to a gated community where sponges and starfish aren't allowed, Squidward ends up trying to escape because he missed Spongebob (and because he was bored of every day being exactly the same).
A memorable example occurs when SpongeBob and Squidward go through hell just to deliver a pizza, and when they finally reach their destination, the customer rudely rejects the pizza all because they forgot to bring him a drink (which he didn't even order), causing SpongeBob to break down in hysterical tears. Squidward weakly tries to cheer him up, but proceeds to angrily march back to the customer and slam the pizza in his face.
Squidward: You know I hate you, right? SpongeBob: Yes. Yes I do. Squidward: Well, I hate the smell of burning Krabby Patties even more. Please come back and be the fry cook again.
Another one-sided case: Vlad Masters and Jack Fenton of Danny Phantom have this... Jack believes that Vlad's his "best bud" but Vlad actively wants to kill Jack. However, Vlad doesn't do much to disillusion the man, as Jack is an idiot savant and not even having not spoken to the man in twenty years seems to have dissuaded him from it. His put downs range from slamming the door on Jack after inviting the rest of his family into his home to "volunteering" Jack for a suicide mission. Sad, considering they really were best friendsbefore the accident. And Jack never caught onto Vlad's hatred of him until the series finale, where Vlad finally spells it out for him.
Skwisgaar and Toki from Metalocalypse. Skwisgaar almost never speaks to Toki except to insult him, yet Toki seems to hero-worship the guy. A few episodes even imply Toki has a crush on him.
To the point that in the Season Finale of the first season, when they are about to get killed together, Toki says he always hated Skwisgaar, and Skwisgaar responds with the words "I knows, Toki, I knows". They are said in a loving tone instead of a hateful one, practically proving this theory right.
Also of note, in the episode "Tributeklok", Toki is revealed to be a member of the Dethklok tribute band 'Thunderhorse', as Skwisgaar. He explains: "I'm not payings tributes to myself because in Thunderhorse I'm Skwisgaars!"
Hell, the entire band is pretty much like this with each other most of the time (when they're not hiding the fact that they really do care about each other), with Skwisgaar and Toki being the best in-group example.
In an episode of The Fairly OddParents, Jorgen calls Cosmo and Wanda his two closest friends, which completely surprises them since in nearly every appearance he's made before and since he's spent it berating them and putting them through painful ordeals. He's also friends with Binky, which usually ends painfully for him. Being Jorgen's friend is generally painful.
Jorgen: I don't have many friends, okay? I mean, you punch them in the face once and it's all, "Ooh, stop punching me!"
She says as much in "The Ember Island Players", when she gives Zuko advice and then punches him.
Zuko: Ow! What was that for? Toph: That's just how I show affection.
In a way, Zuko and Iroh. It's clear that the two of them care for each other, but Zuko frequently acts insulting, domineering, and snappish toward his uncle, while his uncle is seemingly oblivious. On the flipside, Iroh more or less constantly trolls his nephew.
After Iroh devotes nearly an entire episode to interrupting Zuko's search for the Avatar solely to buy a new lotus tile to replace the one he lost:
Iroh: Prince Zuko, you're really going to get a kick out of this. The lotus tile was in my sleeve THE WHOLE TIME! *shit-eating grin*
Which becomes some Fridge Brilliance later on when it becomes clear that Iroh is a member of the underground White Lotus organization, which uses the white lotus tile as a secret handshake...
After his Heel–Face Turn, Zuko with Aang. They do get on each others' nerves frequently, and Aang's idealistic views are often made fun of by Zuko, who is more glass-half-empty. However, they learn to appreciate their differences and are able to accomplish many in doing so.
The show has quite a few including Black Canary and Huntress, Hawkgirl and Wonder Woman, and the Flash and... anyone. As Hawkgirl described her relationship, "We're like oil and vinegar. We go together, but we don't mix."
Occasionally, Popeye and Bluto were depicted this way. The best example is "Fightin' Pals", which starts off with the two of them exchanging "playful" punches just as Bluto is heading off to explore Africa. When months pass and Bluto doesn't return, Popeye heads to "Darkesk Afrikey" to find his old pal. Driving himself to exhaustion, he finds Bluto... surrounded by lovely native girls, living in a tropical paradise. When Bluto sees Popeye on the brink of death, he hauls out a can of spinach and revives him. The two exchange polite greetings... and proceed to start beating the living crap out of each other, laughing all the way. The weirdest Moment of Heartwarming you'll ever see.
An Al Brodax Popeye cartoon had Popeye and Brutus starting out fighting over who will take Olive's picture for a fashion magazine cover. While they fight, Alice the Goon shows up and paints a very abstract picture of Olive, who loves it. Popeye and Brutus see it and start laughing at it. Olive smahes the picture over their heads, calling them uncouth. As they leave:
Brutus: I'm not uncouth. I'm couth!
Popeye: So am I. Real couth!
Both: ( to the last line of Popeye's theme) 'Cause we're real couth fighting men!
The Tallests of Invader ZIM have something like this, constantly bickering and making jabs at each other. It's never explicitly stated what their relationship is, only that they became Tallest at the same time because they happen to be approximately the same height. But they do visibly dislike each other, even though they're together all the time and are often equally flamboyant and nasty.
Homer and Ned in The Simpsons. Homer insults Ned, steals from him (even to the point of breaking into his house!) and continually wishes him harm. Yet, when it comes to helping Ned through his anger issues, rebuilding his house after a hurricane, saving his business by drawing in potential customers, helping him find a new wife or showing him how to have a good time, Homer is always there for him.
Homer: Stupid best friend Flanders.
Brainy and Clumsy are like this on The Smurfs; Brainy really isn't any nicer to Clumsy than he is to anyone else, but for some reason, Clumsy still considers him his best friend, and is willing to stick his neck out for him in the worst of times. In fact, Brainy does sometimes wonder just what he ever did to earn such loyalty from him, as proven from one episode where Clumsy saves him from what Brainy calls "the second worst situation they've ever been in." (Naturally, this causes Clumsy to ask what the worst was, and Brainy recalls an incident the previous summer with Azrael... And guess who shows upfive seconds later?)
Buck and Larry from Time Squad. Larry frequently is very kind to Buck with his actions (much less with his words), cooking him meals, cleaning his weapons, sewing clothes for him and, according to Otto, disinfecting his underwear. Buck usually shows no gratitude whatsoever - in one episode, after exclaiming that "since Larry cooked us such a tasty dinner, I'll now amuse him with some anecdotes" takes out a diary he stole from Larry's room. However, Larry isn't blindly devoted to Buck whatever venom he spits at him; they regularly participate in old-married-couple-like quarrels, mainly throwing insults at each other, though sometimes Larry complains about being overworked and underappreciated. Usually Buck will reply telling him that he's just a robot and he's supposed to do what robots do. Larry then goes on about how he can't believe he is just a robot to Buck, and how he has dreams, needs and feelings too. But they have moments of Aww, Look! They Really Do Love Each Other (usually then spoiled by Buck, but not always) and blatant Homoerotic Subtext.
This was the case between Darkwing Duck and his next door neighbor Herb Muddlefoot. Herb is usually too friendly and Darkwing found it annoying most of the time, although there were a few times when he stuck up for the guy. (Biggest example: an episode where Herb proved himself SO dedicated to his job - selling kitchen storage products - that he stood up to the villain who was trying to ruin the company's good name. You have to admire someone like that.)
Darkwing has a similarly one-sided relationship to his sidekick Launchpad. Launchpad, being a Cloud Cuckoolander, gets on Darkwing's nerves a lot, but nobody is going to get away with messing with him. Launchpad rarely minds Darkwing's insults and is fiercely loyal to him.
On King of the Hill, this is Hank Hill's relationship with...well, pretty much every character who could reasonably be called a friend. No matter how much they piss him off, he does whatever he can to keep them out of trouble, and there's no one they'd rather have looking out for them.
Played straightest with Kahn Souphanousinphone, Hank's Laotian neighbor. Kahn does little but insult Hank and his friends as uncouth hillbillies, but occasionally shows that he likes and respects Hank. He even calls Hank his best friend in one episode, to the latter's astonishment.
Lampshaded in "Road to the Multiverse" when Brian and Stewie go to a version of Quahog where the US never dropped the atomic bomb on Hiroshima and Japan won World War II, resulting in them annexing the US:
Japanese Stewie: I no like you, but I like you! Japanese Brian: I no like you, but I like you, too!
Applejack and Rainbow Dash are regularly downright friendly with each other due to their similar tomboyish styles, but their competitive and prideful natures sometimes pushes them into this kind of relationship.
AJ also has shades of this in her relationship with Rarity — they'll regularly argue, mock each other and bring up each other's most embarrassing moments, but if one of them needs help with something important, the other is there.
Twilight has shades of this with Rainbow Dash as well. Given that Twilight is probably the most deadpan of the snarkers in their group, and Rainbow Dash does the most of the things done without thinking (not to mention the obvious nerd-jock stigma), they both like to rib on each other from time to time. However, both of them respect each other and their abilities (Rainbow even said she thinks Twilight's awesome during "Castle Sweet Castle") and they know they can depend on each other in a crisis.
Baljeet and Buford in Phineas and Ferb. They even sing a musical number (called "Frenemies") about their relationship in one episode. There is also an entire episode about Baljeet and Buford "breaking up" after Baljeet realizes Buford could track him by a GPS, which proves to be Baljeet's last straw, which causes Buford to work with Doofenshmirtz. It is revealed that Baljeet and Buford do like each other, as if they're in a relationship.
Daria and Jane, which become more blatant as the series progresses.
Archer and Lana, possibly also his mother. He really does care about them, and they feel the same way, even if they'll never admit it and sometimes want each other dead.
Pam and Cheryl are a great example: Pam didn't seem to care when she thought she murdered Cheryl, and once told her to kill herself as part of a tontine. They both openly insult each other and Cheryl has gleefully framed Pam to cover her own ass. Despite all of this, they're two of the only characters to hang out outside of work, and even seem to live near each other.
G.I. Joe had Leatherneck, a Marine and Wetsuit, a Navy S.E.A.L. They're constantly praising their own branches while snarking at the other, have come to literal blows over the slightest thing, but undeniably have each other's back such as Leatherneck getting poisoned by Serpentor where Wetsuit let himself get bit by Serpentor's pet to get an antidote. Upon recovering they returned to insulting each other as a sign of things being ok.
Regular Show: The main duo with Muscle Man and High-Five Ghost. Though Muscle Man constantly makes jokes and pranks them at Mordecai and Rigby's expense, they gradually hang out more and more due to their similar interests.
Also, Mordecai and Rigby themselves.
Bender and Leela in Futurama. The only thing preventing the two from leaving each other stranded to die a horrible interplanetary death are the two's feelings for Fry by proxy. That and keeping their jobs.
One of the later episodes reveals that Zoidberg thinks that he has this kind of relationship with Hermes (being Zoidberg, he is wrong about this).
Miss Finster and Miss Grotke in Recess. The latter thinks of the former as her friend, though most of the time she's shown not to like her that much. Though there's the occasional Pet the Dog moment between the two of them.
Squirt and Niblet from The Hub's Pound Puppies. They usually get along fine and interact like good friends. But the episode "Homeward Pound" tested their friendship when they ended up lost in the Canadian wilderness. Squirt took charge and ordered Niblet to follow, but soon, they started to argue and split up. They ended up rekindling their bonds at the end, though.
Roy Rooster and Wade Duck from 'U.S. Acres/Orson's Farm' in Garfield and Friends. Although Roy's pretty much this to the rest of the cast, his friendship with Wade fits this trope much more (and more in the cartoon version.) He's always making pranks to scare Wade and other times they argue but there are other times where they act like real friends. Over the seasons their love/hate friendship evolved to the point where they had the closest friendship with each other, besides with Orson.
Goofy and Pete on Goof Troop are a one-sided variety, with Pete being actively antagonistic or purposely using Goofy while Goofy remains blissfully oblivious to the fact that Pete is a jerk who doesn't like him and considers him his best friend. There are a few hints that Goofy does in fact realize that Pete is a jerk and just doesn't mind, sometimes Pete admits that he does actually sort of like Goofy, and their relationship is much better in the movies (except for Pete's parenting advice).
Various engines of Thomas the Tank Engine are known to have their quarrels and laugh at each other's misfortunes, but when it comes down to it, they are still shown that they are good friends and help each other out no matter what.
Gordon and James would often bicker and tease each other for their mishaps, although this is only because the two have equally haughty attitudes. However, they are still seen hanging out with each other along with Henry in their Freudian Trio. Henry is also this towards them, too.
Let's just say James has this sort of relationship with a lot of engines.
Thomas and Percy have their moments of arguing and teasing each other since both tank engines share the same cheekiness and mischievous nature, but aside from that, the two always look out for each other and are really the closest friends of the series.
Frizz and Nug are often depicted as this in The Dreamstone, they bicker childishly on a frequent basis, but are pretty united about their contempt for their line of work and superiors such as Urpgor. Rufus and Amberley also have moments of this, albeit to a much lesser degree.
Steven regards Lars, a snarky, surly teenager who works at Beach City's local donut shop, as "basically my BFF", even when Lars is teasing him about having a crush on Connie in "Bubble Buddies". On a rare few occasions, Lars seems to reciprocate that friendship, or at the very least appreciates that Steven considers him a friend.
Pearl and Amethyst squabble frequently, but they're quick to put aside their differences and work together to accomplish their missions or help Steven, and there's even signs that they really do love one another as family (such as Pearl giving Amethyst a comforting hug in "On the Run").
On the rare occasions they are separated, Ruby and Sapphire seem to have shades of this trope, although that may be because they mostly only separate whilst in an argument. Additionally, they are in a romantic relationship, so slightly more than best buds
The titular chipmunks in Chip 'n Dale Rescue Rangers are like this. Many episodes include at least one heated argument between them, usually triggered by the fact that Dale has said or done something foolish, prompting the more serious-minded Chip to berate him. These quarrels often involve Chip bonking Dale on the head or kicking him up the backside. However, the pair are close friends for all their squabbling, as Chip's reaction when Dale is captured by Fat Cat's gang as the result of a prank war taken too far illustrates.
Rugrats — Angelica with the rest of the kids on the show, especially with Susie Carmichael. However, given Angelica's mean and bossy personality, Susie's probably the closest thing she has to an actual friend outside of Tommy, Chuckie, Kimi and the DeVille twins.
Noah behaves this way towards Cody, Eva, Izzy, Owen, Katie, and Sadie. If he's given someone an insulting nickname, chances are they've somehow made it into his good books.
Chris and Chef bicker constantly but are still partners-in-crime who love making the contestants miserable.
The Ridonculous Race gives us Gerry and Pete, two lifelong tennis rivals who constantly insult and sabotage each other but are still fast friends.
Gerry: We're both very competitive, but for half a million each? Game on! Pete: Maybe we'll even get some new sponsorships. If anyone needs a pitchman for senior laxatives... Both: Call his agent!
It depends on the writers but on default The Misfits from Jem are this, at worst they only seem to be together because they're bandmates. They constantly bicker and fight but several episodes show friendship between them, especially Stormer and Roxy.
Dan and Chris from Dan Vs. are like this, which is probably the closest Dan can get to actually being nice to someone.
The eponymous Wander of Wander over Yonder is friendly and nice to just about everyone... even people who seem to hate his guts, especially Lord Hater.
A good number of episodes of The 7D focus on the dynamic between Grumpy and Happy. At worst, the former finds the latter an intrusive nuisance who sings all the time. Otherwise, they get along pretty well and even have a few things in common.