Dragon Ball Z: Goku has several of these such as Vegeta and Krillin. Their initial relationship was, much like many of Goku's friends, fairly antagonistic.
Also, Trunks and Goten, at least when they're 8 and 7 respectively. Trunks calls Goten an "idiot" or a "little child" numerous times, but with a big grin on his face and clearly no malice; less commonly, Goten calls Trunks out on the fact that he's a member of the world's richest family and has no real reason to be competing for money like he does. Once they become teenagers, this snarking has largely faded, to be replaced by Homoerotic Subtext.
Yamcha and Tien occasionally gave out hints of this, as well. They start out as pretty bitter rivals (no doubt accentuated by Tien breaking Yamcha's leg out at the end of their tournament match), but after Tien's Heel-Face Turn, they become pretty solid friends, if occasionally still snarky toward one another. Yamcha even jokingly calls Tien " a little pig-headed" before the latter makes an attempt to help Gohan by blasting Cell... which is followed immediately by Yamcha joining him in the attack.
Bulma and Chi-Chi in Z. They both find themselves arguing over their husbands and sons, but they do get along most of the time.
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.
Bleach: Almost all of Ichigo's closest relationships are based on this trope with the exception of his relationships with Sado and Orihime.
Ichigo and Rukia argue a lot. Rukia will punch Ichigo and shout if it means restoring his resolve, and Ichigo will snark about her attitude and her bizarre taste in art.
Ichigo and Uryuu began as snarking enemies but very quickly faded into not-friends who snarked each other constantly and pretended to be enemies. They still can't admit they're friends but their classmates and the rest of their group find their vitriolic relationship to be very amusing, especially given how alike the two really are. Orihime has begun lampshading how strong their friendship has become, much to their horror. Uryuu in particular hates having this pointed out. It's also a Generation Xerox example - their fathers have almost the same relationship with each other.
Ichigo and Renji also tend to butt heads within moments of trying to do almost anything, often to the point of immediate physical violence, but will nonetheless put their lives on the line for each other at a moment's notice.
Ichigo and Tatsuki have a violent, argumentative relationship that goes back to their distant childhood.
Kurumu, of Rosario + Vampire, becomes this kind of friend towards Mizore and Yukari after character development sets in, teasing them about the weirdness of their respective species and their attempts to snare Tsukune. (The manga is predominately a harem series, and all three originally met because they were interested in the same boy.) It's a mark of how far their characters have evolved when they can let go of their jealousy and fear to snark with the playfulness that they do, and the manga has no shortage of moments when they drop even that and just support each other.
Gintoki and Katsura have their verbal scuffles, the latter tries to drag the former back into their resistance group while the former tells him in so little words to shove it — but when they team up against an enemy or twelve, the results are frequently awesome.
The Shinsengumi have a bond similar to this, as well. They even refer to it as a "rotten friendship".
From Shaman King we have Yoh Asakura and Ren Tao with the latter usually snarking at Yoh's ideals and trust in others. Though Ren will usually go along with him anyway.
This could also apply for Ren with both Horohoro and Chocolove as well.
Maka and Soul in Soul Eater. Soul heaps verbal abuse at his meister and gainsays her half the time, while Maka usually replies with a heavy book to Soul's face. Naturally, the two are inseparable and will Take The Bullet for each other without a moment's hesitation.
In the first episode it appears Soul does not even like Maka that much... until:
Later episodes and chapters make it known that this is very much a close relationship and some chapters seem to imply that their snark has to do with their own insecurities (like Soul's inferiority to his brother Wes and Maka's inferiority toward other[female] meisters who want to be Soul's partner).
Maka and Black*Star also fit this. Maka doesn't let Black*Star get away with his arrogance quite as much as Tsubaki and Soul seem to, and Black*Star tends to call her out on her own faults...but they've known each other since they were small (Unlucky Childhood Friend to Soul/Tsubaki, maybe?) and Black*Star not only hunts down the person who left Maka paralyzed, but becomes extremely angry with and attempts to fight Crona when Crona claims to have no memory of Maka.
Rosette and Satella personify this too, almost to a greater extent.
Both Digimon Adventure (Taichi/Tai and Yamato/Matt) and Digimon Frontier (Takuya and Kouji) had a form of this with their main characters and lancers, of the both insulting each other variety. Ruki and Takato, from Digimon Tamers, were less best friends, since theirs was a Power Trio situation, but it's still a similar outcome to the first description.
The Straw Hats from One Piece in general are like this toward one another, at any given time at least two or three of them are arguing and/or hitting each other. They're basically a Vitriolic Crew. But don't let that fool you, they're all immensely fond of each other. If you hurt one of them, the others WILL kick your ass (and considering they are, as of now, considered one of the most dangerous crews on the Grand Line...). The most notable example are:
Sanji often expresses disdain towards the male members of the Straw Hats, usually only seeming to care about the safety and well-being of the female members (all two of them). He has shown to care for them, particularly Usopp, he just won't express it. This was also his relationship with his adopted father figure.
Some of the loudest shouting matches are between Nami and Usopp, with either one calling out the other on whatever zaniness they're trying to pull. When they're not fighting, however, they're working together to keep up with the physically-stronger members of the crew. Usopp even provides Nami with magic weapons and helps her work out strategies for using them. This is probably because Nami and Usopp are the only members of the original crew with above-average intelligence (Chopper, Nico Robin, and Franky have not reached this level of closeness).
Also outside the crew, the way Ace and Sabo treat Luffy initially seems to be this.
Iceburg and Franky who argue about everything, from when they were young to even when they were adults. But when Iceburg discovers that Franky wasn't actually dead after being run over by a sea train, he has this to say.
Iceburg: "We never change, do we? We yell at each other as soon as ... look, I won't forgive you and you probably won't forgive me either, but Franky..." (tears up) "... Idiot ... I'm glad ... that you're alive!"
The relationship of Shanks and Buggy continues to be this, no matter how much time has passed, the huge gap between their abilities, or how much Buggy continues to deny it.
Just about every word Sawamura says to Rintarou in Wa Ga Na Wa Umishi is snarky. He's not above hitting him either. Yet somehow the two work together great when a job needs to be done, right down to saving each others' lives multiple times.
Keroro and Giroro from Keroro Gunsou qualify. These two, despite their constant arguments, are shown to be extremely close and will always pull through for each other.
Most incarnations of Sailor Moon have this in some form:
Rei and Usagi in the anime version fall into a softened version. Usagi does get upset by Rei's name calling but Rei is her closest friend out of the Sailor Senshi (her death affects Usagi much worse than that of the other girls) and Rei seems to be trying to toughen Usagi up and it is Rei who she entrusts with the silver crystal when she can't trust herself not to give it to the enemy in return for Mamoru, and later is the one who expresses more sympathy for Usagi when she finds out in Stars that Mamoru has disappeared and Usagi was a Stepford Smiler over it.
The DiC dub made it seem Raye was trying to usurp Serena's position as leader at times instead due to removal of subtlety in the dub. They blatantly made Raye openly express a desire to be leader instead of Serena and changed the show of trust between the two surrounding the silver crystal into having Raye keep it when Serena left it at Raye's house by accident instead of returning it.
The anime also, at times, had this between Minako and Makoto. Usually when they were arguing over a guy. Interestingly this showed up in both their second season and fourth season power up episodes.
In the Anime in particular Haruka is always the harshest on Usagi, calling her out on her hopeless idealism, getting frustrated with her tactics or lack there of, and even violently jumping her at one point just because she was angry at the way Usagi handled a situation. Where as Usagi doesn't appreciate Haruka's rough manner of dealing with things and close-mindedness. But deep down it's shown that Haruka has an incredible amount of love towards Usagi. Feeling a fierce protectiveness for her and getting incredibly angry when anyone even indicates that they might harm her. Likewise, Usagi greatly respects and looks up to Haruka, and (though she doesn't always trust her judgement) she does appreciate Haruka's concern.
In every adaptation Chibiusa's in she has this kind of relationship with Usagi. It makes this trope because the two aren't technically sister like they pretend to be and Chibiusa doesn't yet consider Usagi to be her mother They often fight over treats, toys, and Mamoru's attention. Chibiusa complains about Usagi's immaturity while Usagi complains about Chibiusa's cruelty and general brattiness. But the two do love each other very much. They both are genuinely sad when ever Chibiusa has to return home, and Chibiusa draws a picture of Usagi secretly naming it "the person I admire most" Then of course you have Usagi risking her life to save Chibiusa's best friend in the S season, jumping off a platform thousands of feet in the air in order to save Chibiusa in Super S, and jumping into a black hole to save Chibiusa in the Super S movie
In the manga, Rei has this type of relationship more with Minako, not Usagi.
The live action version followed the manga in the case of Rei and Minako's relationship with hints that there may have been something else hidden under the fighting to the point that during fights the two always had each others backs and when Minako died it was Rei who was most upset out of the senshi despite how only a few episodes earlier Rei left them due to a fight with Minako
Essentially the relationship between Saki and the entire Genshiken through most of the story. Saki is disturbed or confused about the nerdy (and porn-obsessed) interests of the rest of the club, and responds to them with snark and insults unto the very end, despite eventually regarding most of the club members as close friends. Also, although she claims to wish for the day the Genshiken club is closed, she's more than willing to stand up for them.
Crest of the Stars: Given how large part of their social life is based on a culture of polite insult, it's not surprising that most of the Abh friendships happen this way. Among the best examples are Bibauth brothers, Crown Prince Dusanh and his Chief of Staff, and Admiral Spaurh with, well, everyone.
Jerk with a Heart of Gold Edward Elric of Fullmetal Alchemist this kind of relationship with his superior Roy Mustang. He complains about Mustang piling on orders, and the two snark at each other, but in reality, the two care about each other.
Ed's relationship with Ling. Ling is sort of an Idiot Hero (or at least acts the part) and considers Ed a good friend, Ed is annoyed with Ling's mooching food and money off of him, seeking immortality — essentially an evil goal given the method of getting it in this universe, and for being much taller than him even though he's a year younger. Deep down, Ed does actually like Ling (which becomes obvious when the two are trapped in Gluttony's stomach).
Konata and Kagami from Lucky Star are pretty good examples of this (or Belligerent Sexual Tension, depending on how seriously you take the Les Yay vibes). Much of their dialogue consists of verbal jousting, and the barbs get quite vicious on occasion. You just know that both would happily give every organ in their body to the other if they had to.
To varying degrees, Naruto has this with the some of the Konoha 11, most notably Kiba, Ino, Shino, and Neji. Naruto often makes fun of his friends for their personality quirks, and they make fun of them for his, but they stand together in dangerous times, and Naruto's friends don't seem at all bothered by him being host to the demon fox. In a filler arc, the Arc Villainwhom Naruto had often talked with reminds him of how he often complained about his friends, Naruto responds that while they fight, he likes them and trusts them to help stop his Evil Plan.
Revy and Eda from Black Lagoon are either this or With Friends Like These...... It's kind of hard to tell. They certainly haven't done anything selfless to help each other yet, but seem to get along well enough when they share a common goal. (i.e. money, booze, Seinfeldian Conversation, shooting at other people, or any combination of the above.)
Lelouch Lamperouge and Suzaku Kururugi in the Code Geass Sound Stages, when Lelouch and Nunnally were staying with Suzaku prior to the invasion of Japan. They initially hated each other, but became friends after looking for a lost and injured Nunnally together. The two were willing to make fun of the other's physical and mental strengths and weaknesses, but they worked well together and formed a bond before Britannia invaded.
Also Kallen and C.C., particularly in the radio dramas set between seasons, and several scenes in the manga.
Azumanga Daioh has Yomi and Tomo, and Nyamo-sensei and Yukari-sensei. While both of these pairs are clearly adversarial, they do maintain a friendship of a sort. Although in both cases, particularly the latter, it may simply be a case of having been together so long, they're probably just stuck with each other.
There's also England and Australia, based on their Real Life relationship. And Australia and New Zealand.
Greece and Turkey, of all nations, seem to be set up as this in the new character profiles. Apparently, in spite of them taking any and every opportunity to exchange blows with one another, fighting over Japan's attention, and making England and France look positively peaceful in comparison, Greece still goes shopping in Turkey's malls.
The Frying Pangle are also shown to have elements of this, especially if it involves Prussia.
Let's not forget the title character and Koga from InuYasha who, as romantic rivals, could, according to Kagome, "scrap (...) all day"; have fought side-by-side against numerous adversaries while trading barbs about each other's ancestry, personal cleanliness and fighting experience. (Or, rather, the lack thereof...). The rivalry-friendship evolves notably from the first appearance of Koga as a main sort-of villain, when he is quite determined to kill Inuyasha in a No-Holds-Barred Beatdown, and it isn't even remotely funny. Once he discovers that Inuyasha wasn't the one who killed all his friends, he settles into a comfortable role of showing up occasionally and being annoying. In fact, the two have a tendency to rescue each other from various horrible situations, snarking at each other all the while for being unable to escape themselves, and they keep it up right until the end, when Koga's Jewel shards are stolen, but there is definite friendship between the two. Which is possibly why so many fans were up in arms when Inuyasha stated in the final battle, "there isn't one missing," and Koga notably was.. In fact, some fans seem to have realized that theirs is perhaps the most ultimate of Rumiko Takahashi's slashes
Ash and Misty from Pokémon practically named this trope. This is due to the fact despite being friends they are rivals for being top Pokémon trainers, and because their personalities naturally clash with each other. Many characters seem to take their bickering as a clue to "Aww, Look! They Really Do Love Each Other".
Nurse Joy: What's the saying? You only hurt the ones you love?
Ash and Misty: (both blush) No way!!!
Jessie and James in Team Rocket. Though they are constantly bickering (Jessie used to beat up James quite often) they do genuinely care about each other, are good friends, and remain a team despite constantly losing to a bunch of kids.
Add Meowth. As demonstrated most recently in Unlocking the Red Chain of Events, any one of them will go to the mat for the other(s).
Tepig and Oshawott are a rare Pokemon example. While there have been Pokemon in the past that fought with each other despite being on the same team, these two seem to be the most recurring example. They often fight for juvenile reasons, but most of the time they're happy being on the same page.
Watanuki and Doumeki from ×××HOLiC. Doumeki acts like an ass around him? Check. Will Doumeki spend hours digging under the rain to rescue him? Definite check!
It is CLAMP, though. There's definitely something more on Doumeki's side.
Shinra and Shizuo from Durarara!!. Shizuo spends about half of the friendship (unsuccessfully) trying to resist the urge to beat Shinra senseless (largely for being such a gossip and motormouth). Luckily for him, Shinra has something of a mile-long masochism streak, and doesn't consider things like "he regularly dislocates your jaw" to be barriers to friendship.
Also Light and L fall into this, especially during the Yotsuba arc when Light becomes a Nice Guy that genuinely likes L but L isn't too enthusiastic about it, knowing (but unable to prove) that Light is an (amnesiac) mass murderer. It becomes more of a Type 1.
Matsuda and Aizawa. The latter would criticize the former for his idiocy, but Matsuda still looks up to him.
From Monster Rancher, we have Tiger and Hare, both who love nothing more than to snark and insult each other. But if the other was in any trouble, the other won't hesitate for a second to jump in and help them.
Shinagawa and Izumi of Flunk Punk Rumble. While they spend much of their time together either viciously staring each other down or outright brawling, they make a good team, and in spite of the fact that they clearly don't get along and often outright detest each other, they both seem content to admit that just because they hate each other doesn't mean they don't like each other, and would each inarguably (and demonstrably) go to the ends of the earth to save the other from danger.
Asuka Masamune and Hajime Tonomine from Otomen. Hajime considers Asuka his number one rival and takes every opportunity they meet to see who is better at the task at hand. However Asuka often asks Hajime for favors, and they keep the other's secret, i.e., their Otomen identities. Recent volumes have even shown Hajime having a grudging respect for Asuka, though that doesn't stop him from engaging his rival in battle!
Mitsuba Marui & Miku Sugisaki from Mitsudomoe are generally rivals for the title of the Alpha Bitch. However, if one thinks the other's in genuine trouble (Mitsuba is sick, or Mitsuba thinks Miku's been kidnapped) they'll go to the other's aid.
Natsu Dragneel and Gray Fullbuster of Fairy Tail. They constantly snark and insult each other at every time, however both have proven time and time again that they would sooner take a bullet (or whatever other magical attack comes their way) through the face for the other.
Also Natsu and Gajeel have become this over time, with Gajeel being a bit more snarky than Natsu is, however both of them are friendly to each other in their own way, make a perfectly awesome team, and do actually care about each other as friends.
Really, this could apply to everyone in the Fairy Tail guild; the brawls that spark up between members for no good reason have been legendary and frequent. Often whenever anyone returns from a mission.
Touma and Misaka in A Certain Magical Index are very much this, seeing as most times they talk to each other there are usually some jabs mixed in (unless its a serious situation). it get funny during a couple times Touma calls and asks Misaka for help with something... like getting through a shuttered subway entrance...
Akina and Kyosuke from Yozakura Quartet fight. A lot. Unless they have a common enemy.
CLANNAD: Despite the fact that Tomoya regularly pulls pranks and snarks at Sunohara, they share a close friendship. In actually serious situations, they will help each other.
Mai, Yuuko, and Mio from Nichijou. Surprisingly, Nano is excluded.
Slam Dunk: Akagi and Sakuragi and to an extent, all of Shohoku.
The eponymous Tiger & Bunny don't reaaaaally fit this description until episode 25, when they're working together as willing partners but also squabbling over who's going to pay for a damaged car. Before this they were completely unwilling partners, or Barnaby was busy forgetting who Kotetsu was and hunting him down for murder... Yeah, it's complicated.
Hal and Kai from Switch!. Hal is constantly referring to Kai as 'useless' and 'no good', sometimes even using the phrases in place of Kai's name. Despite this, Kai remains affectionate and loyal to Hal.
Sonic and Knuckles in Sonic X and also Amy and Knuckles to a lesser extent.
One-sided variations appear very commonly in Anpanman. Some characters (normally girls), become friends with Baikinman, despite him wanting to have nothing to do with them. Multiple times, a single kind deed will make the other party believe that Baikinman's not a villain whatsoever, much to the surprise of everyone else, and even when being told the truth, they'll still see Baikinman as a friend, even when he himself tells them otherwise. Even Anpanman himself treats Baikinman with the same kindness he shows others (most noticable in the educational videos, where the feeling reciprocated), but they will go back to being enemies as soon as Baikinman does something to endanger someone else.
Also in the series is the Donburiman Trio. Multiple episodes are made up of them fighting over which donburi is the best, them leaving in a huff, getting upset about being without the others, and reconciling by the end of the episode. Why do they stick together? Because they are the Donburiman Trio, so they'll always have each others' backs and still be best friends, even if they all hate each other sometimes.
Tomoe and Mizuki, Nanami's two servants, from Kamisama Kiss. Mizuki first showed up in a minor villain role before being defeated by Tomoe. A short while later Mizuki entered into a magic contract with Nanami, much to Tomoe's displeasure. They work together and even have a somewhat friendly understanding while at the same time competing enthusiastically for Nanami's affection and approval.
Rei "Hana-no-Saint-Juste" Asaka and Kaoru "Kaoru-no-Kimi" Orihara in Oniisama e.... When they're in good terms, they playfully insult each other while smiling. When not, Kaoru will do her best to support the very emotionally unstable Rei no matter what.
My Bride Is a Mermaid: Kai and Nagasumi spend about 90% of their shared screentime fighting or otherwise being antagonistic towards each other, but during an episode where Kai is Mistaken for Dying Nagasumi is genuinely grieved at the thought and treats him with respect, trying to make his last days as good as possible.
Levi and Hanji from Attack on Titan. They tease each other almost constantly, but they seem to be very comfortable around one another, and aninterview has mentioned Hanji, Levi, and Erwin all having a close, trusting bond.
First there's hikiNEET Shintarou and Ene, the cyber girl who lives in his electronics. They're almost always fighting, and Ene has practically made it her job to troll him whenever chance she gets. Despite this they actually do care about each other and Ene is genuinely trying too get Shintarou to lighten up the only way she can.
Kido and Kano. Kano teases Kido and purposely does things to provoke her (sometimes to points of violence) and the two bicker constantly. They actually care a whole lot for each other, having grown up together and helped the other maintain their eye abilities- crosses over into Slap-Slap-Kiss territory sometimes, as it's hard to tell just how close they are.
Hibiya and Momo have this going on too, even once Momo managed to become a Cool Big Sis to Hibiya.