Doumeki and Watanuki of ×××HOLiC behave like this around each other, much to the amusement of Yuuko (and the readers).
Levi and Hanji from Attack on Titan seem to have this going on. They're implied to have known each other for quite some time and they're almost always teasing one another. (Something most people would get roundhouse kicked for if they tried that with Levi.)
After consummating their love for one another Guts and Casca from Berserk begin acting like this toward one another throughout the entire Griffith rescue operation, with them both getting miffed and jealous toward one another, but are clearly seen being very caring and sentimental toward each other (with some Slap-Slap-Kiss action in-between, especially from Casca). Judeau and Pippin take this as a good thing and play it for the laughs. Griffith, judging from they way he is staring at the couple from underneath his helmet and seeing that he had been through the hideous ordeal of being tortured for a year only to come back to see a change in authority, power, and affection between Guts, Casca, and himselfdoesn't seem very pleased with this...
In the dub of Bakuten Shoot Beyblade Max and Rei describe Takao and Kai this way. The amount of Slash Fic those two inspired back in the day makes this about ten times funnier.
In Brave10, Rokuro acts the nagging wife to Yukimura's lazy lech of a husband. Taken up a notch during the exile arc of the sequel, where Rokuro starts dressing like a housewife and beating Yukimura with a broom for being a drunk who's in the way of his cleaning.
Lelouch and C.C. from Code Geass have a constant back and forth going on, usually involving C.C. stealing his credit cards to order ridiculous amounts of pizza among other things. In addition, they share Lelouch's room, possibly his bed, and C.C. is constantly seen wearing his clothes.
Lelouch and Suzaku also have quite a bit of this, especially when they were kids, and in the audio dramas for both pairs.
In episode 2 of Digimon Savers, Marcus is trying to tell Yoshi to leave, but she refuses, so they get into an argument. Then Kristy walks past:
Kristy:[to Marcus and Yoshi] Stop arguing like an old married couple. Marcus and Yoshi:We're not a couple!
While Izaya and Namie's relationship in Durarara!! began with hostility, they maintained a working relationship because of their confidence in the other's skills. Soon, they're comfortable enough to hang out together outside work (e.g. Namie cooking dinner for Izaya, and having meals together). Even though their interactions consist of mostly trying to annoy one another in some way, they probably know each other better than anyone else.
Bulma and Vegeta do this infrequently in some episodes and the movies too. It's pretty funny watching them snip at each other.
Goku and Vegeta also get a good share of bickering in, mostly while exploring Buu's body in Z and during the battle against Omega Shenron in GT.
Free! has Haruka and Makoto. The latter is frequently seen taking care of Haru, saying exactly what he thinks and knowing just how to convince him of something. The occasional bickering, while quite mild, is also present.
Kyo and Yuki are called out on this (word for word) by Shigure in Fruits Basket. They're both displeased with the comment.
Tsubasa and Misaki from Gakuen Alice would often pair up in couple events. They both scoff at the notion that they might be going out, but their peers noted that they're practically married already. They eventually got together for real, though.
Haikyuu!!: The bond and on-court chemistry between Kageyama and Hinata is so palpable that the first time Coach Ukai sees them interacting with each other he assumes they are friends since childhood. Their interactions fall into two categories of either them acknowledging/acting like they are destined-rivals/teammates or constant bickering like an old married couple. It is especially funny when the focus of a scene is on other characters but you can hear them in background arguing about something as trivial as how to say poop.
In Hetalia: Axis Powers there's France and England, who've known each other since the very beginning of time and probably have a thousand romance tropes under their belt. Not to mention, almost every time they're seen together they are bickering.
Highschool of the Dead: It takes awhile for them to get there, but it's clear that Hirano and Saya are made for each other. By chapter 29, it's gone beyond the point where Everyone Can See It, to the two of them practically being Alice's foster parents.
In K, Saruhiko and Misaki, whenever they're not at each others' throats. The audio dramas add a lot more of this between them.
In the Love Hina manga, when Keitaro decides to get a job, he suits up and all the girls laugh at him. Naru however, does up his tie, leading Kitsune to make this exact comment about them. Denials follow.
Lovely Complex: Koizumi and Otani are accused of acting like this - after they're already together, surprisingly, but they still both get embarrassed.
Konata and Kagami from Lucky Star are pretty good examples of this. 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.
In Magical Record Lyrical Nanoha Force, Signum and Agito have a heated argument over Signum's decision to cut her hospital stay short, which ends with a mixture of affection and pouting. Apparently, five years were enough time for them to start acting like a married couple (even though the bond between a Lord and her Unison Device is pretty much just that).
Maoyu: The Hero and the Demon Queen make an arrangement that they will belong to each other, which makes them husband and wife in all but name, and they even act as such most of the time. There's no denying that the two are very attracted to each other, but their interactions tend to lean more to the sweet and innocent side.
Mazinger Z: Kouji and Sayaka were like this. Once while they are arguing -and fighting- in the background, Shiro is fortune-telling. He notes the cards tell his brother and Sayaka are destined to be together forever.
Since the very moment they met, the interactions between Hachiman and Yukino from My Teen Romantic Comedy SNAFU are basically the former making fun of the latter and explaining why his view on life is wrong as he tries to defend himself from her arguments. They have no problem acting like this in front of everyone who, one way or another, conclude that the two get along just fine.
Nadia: The Secret of Blue Water: From the moment Jean and Nadia find Marie, they become an almost familial unit - with the two of them essentially acting as her guardians. Marie even makes comments about how they are acting like a couple, whether they're discussing over whether to help the imprisoned villagers or not, or when they're about to go flying in a gyrocopter.
Even funnier in episode 17 because when Marie teases Nadia about her feelings for Jean, she blushes and denies it. (But she approaches Jean anyway.)
Naruto has one with the villains; both Deidara and Sasori and Kakuzu with Hidan bicker to one another like, as fans put, this trope. However, they take it in two different ways; Deidara and Sasori, being artists, bicker about their type of art (apparently often), even when the situation at hand has nothing to do with it. Kakuzu and Hidan, on the other hand, already hate each other, so all they need is one reason, and they're very likely to go off on one another and spend a pointless amount of time doing so.
Shinji and Asuka. The above Running Gag about Nadia and Jean was parodied by Toji about them in episode 17.
Toji: What's this? Married couple fightin' again?Dub Uh oh. The newlyweds are fighting. (Entire class laughs. Both assume Luminescent Blush.) Shinji & Asuka: It's not like that! Dub Oh, shut up!
Shinji and Asuka double on this score in the Lighter and Softer manga spinoffs. There, Asuka's rage level and Shinji's Extreme Doormat level are both turned down, so there's more arguing and more friendship at the same time. They're always together, care for each other deeply, and... never stop griping at each other. (Add to that Asuka crushing on Shinji, Shinji not noticing, and that being the cause of Asuka's more canon-Asuka-ish moments.) It's still Toji who jokes about it; he refers to them as "the newlyweds" more often than he actually uses their names.
When Kaji arrives late and somewhat disheveled for a wedding, Misato first criticizes him for his tardiness, then for not having shaved recently, before finally correct his loose tie. When Ritsuko comments on this behavior between them, pointing out that they act much like a married couple, Kaji is amused, Misato is not.
Yato and Hiyori act very much like a goofy husband without a job and his nagging wife, with Yukine acting as their surrogate child. They argue a lot, but it's clear how much they care about each other.
Kofuku and her Shinki Daikoku also aren't married, but they surely seem like it since they have been together for centuries and they get along very well despite him getting fed up with all the trouble she causes as a poverty god.
Nami and Zoro from One Piece, almost constantly due to their hotheaded temperament. This happened quite often until the timeskip onwards, when they seem to have fewer interactions between each other.
Gilbert and Alice have the tendency to act like this. All the time. They can hardly have a serious conversation without making some sarcastic retort to the other. Oscar even made the comment that "They get along well enough to bicker".
Elliot and Leo also have the tendency to bicker over various things, and some of these bickerings may escalate to an outright fight with a lot of fighting and throwing of various nearby objects. Despite this (more like because of), they're still best friends.
Patlabor: While the relationship between Captains Gotoh and Shinobu is strictly professional, they fit the trope to a tee. Their bickering rarely goes beyond snarking, and they relate as if they've known each other their entire lives. This is best seen in "The SVU's Longest Day" in the OVA continuity, and the entirety of the "Two in Karuizawa" episode in The Mobile Police/New Files continuity.
Ranma ½: This observation is what gets Ranma and Akane engaged. Even their classmates notice.
"Ooo! A lovers' quarrel!" "Practically married already."
Some of Rei and Usagi's arguments are petty like this in the Sailor Moon anime, whether in or out of their superhero roles. Whenever the friend group meets up at Rei's house, there is a 99% chance that Rei will be telling off Usagi for snooping around her bedroom (and, in one instance, Rei tries to wrestle a comic book out of Usagi's hands in fear that Usagi would get tear stains all over the pages) or behaving "too Usagi" and giving her second-hand embarrassment, but it's clear that the two girls are good friends, respect each other and would help each other out, no questions asked.
Maka and Soul. They are very comfortable around each other, they live together, have the occasional lover's spat and Snark-to-Snark Combat and can read each other well. Arachne even comments on of the fights they had as entertaining.
Black☆Star and Tsubaki are all but Happily Married. They are even raising a kid together!
In Tanaka-kun is Always Listless, Word-for-word, this is Tanaka's picture of his relationship with Ohta and the apparent reason for its stability.
In Teasing Master Takagi-san has an adorable "rivalry" with his cute classmate, with them each trying to get the other to react or blush. They aren't quite at the point of being an Official Couple, but the rest of their class thinks of them as such. They do reach that point, however, as there is a spinoff series set ten years later in which they're married and have a preschool age daughter, Chi.