Bill Dundee and Jerry Lawler, specifically in the state of Memphis. When they weren't teaming they were feuding and when they weren't doing either of that they would still find time to snipe at each other.
It usually can be seen with announcers, in some cases more than others, especially Gorilla Monsoon and Bobby Heenan who delighted in heaping insults on each other much to the fans' and their own amusement. Despite their adversarial and sometimes acrimonious on screen relationship, in real life they were very close friends.
Bret Hart and Curt Hennig. Even as a tag team, when one guy calls himself "Mr. Perfect" and the other "The Greatest There Ever Was, Is, Or Every Will Be", you know there is going to be some vitriol between them.
Jim Cornette and Paul Heyman, the two manager of the Midnight Express, owners of rival companies Smokey Mountain Wrestling and ECW, owner and supervisor respectably for a time of WWE's former developmental territory OVW, one of whom created the three way match, or more likely stole it from someone else and then could not decide whether to take credit for it or give credit to the other before finally admitting the concept is older than their time as bookers.
Konnan has openly criticized the running of WCW, WWF/E and TNA, including Kevin Nash's involvement in all three, yet the two are still civil with one another.
Shawn Michaels and Triple H, both in and out of the ring. When they're faces, the mockery is light-hearted and funny (i.e. only Hunter is allowed to mock Shawn's receding hairline; only Shawn can get away with mocking Hunter's thongs). When they're heels, as Shawn puts it, "You'll let your best friend do things to you in the ring that no one else could get away with."
If one looks at the careers of Rey Mysterio Jr. and Eddie Guerrero up until their first and final meeting, it certainly seems this way with the amount of times they alternate between teaming up together and feuding.
Nobuhiko Takada and The Great Muta have open disdain for one another, or so they claim. It is kind of hard to tell, given the frequency they end up working together or for each other.
Aja Kong and Chaparita ASARI are more of the "best buds" than the vitriolic, being tag team partners, stablemates and all but have a reputation among fans who remember them in states as being enemies thanks to Aja "working too stiff" with ASARI being one of the reasons Kong never again appeared on National Television in the USA.
Lance Storm credits Chris Jericho for keeping him in the wrestling business but since he did Lance Storm would spend much of his time in the business shooting down Chris Jericho's dumb ideas.
The Undertaker and Kane, despite being 'half-brothers'. They've fought as a tag team together and fought each other multiple times. As JBL noted on the reunion match of the Brothers of Destruction, "You mess with Kane, you got Undertaker. You mess with Undertaker, you got Kane." Also on their reunion match, most of that time was shown "one-upping" each other as if they were having fun.
Undertaker has actually said that two of his few joys are fighting against Kane and fighting alongside Kane.
On a promotional level, this was the relationship between ECW and the World Wrestling Federation, as they both saw a common enemy in WCW and were willing to help each other out at times, even as they bashed each other during their shows.
Another promotional level case is between WWC and IWA Puerto Rico. They shared many talents and practically had the same exact women's division. When IWA got big enough to put WWC out of business, those predicting WWC's demise were shocked when owner of IWA Puerto Rico and Japan, Victor Quiñones, loaned WWC money to keep them afloat. Yes, they took shots at each other and scheduled counter shows but the only real divide between them seemed to be that IWA PR owner Savio Vega wanted nothing to do with WWC owner Carlos Colon's friend, Invader #1. And even then, Victor Quiñones overruled Vega and brought in Invader #1 anyway.
The Rock and Mick Foley. They began teaming up to take on common enemies, with Foley being much happier about it than The Rock. The Rock, however, couldn't stand Foley. Later, after Foley was fired by Stephanie and Triple H, he warmed up to Foley, even convincing the entire locker room to walk out if he wasn't reinstated. Al Snow was also this way toward Foley, even getting jealous when Foley and The Rock were teaming up.
Internet writer Scott Keith invented the term "Wacky Tag Partners Who Hate Each Other" to describe this phenomena.
AJ Styles and Christopher Daniels have either tried to kick each other's heads in over envy and pride or stood up for each other against TNA's enemies so many times they have to be this. In real life, they've named their children after each other.
Samoa Joe and CM Punk. They don't just feud across three different continents over wrestling titles. Joe doesn't just serve as Punk's "unstoppable monster Of The New Dawn". Samoa Joe also apparently loves women on top of CM Punk's furniture.
Slyck Wagner Brown and April Hunter during their run as Jersey All Pro Wrestling tag team champions. Besides professing the two had a mutual desire to kill each other on the road trips, they were also mostly apathetic to whatever problems the other might have been having with the rest of the locker room, sometimes actively instigating issues against the other. Should one of them come to any serious trouble though and they'd quickly show why they were the tag team champions. April also stated the vitriol was beneficial in at least one instance when Slyck was the only wrestler present willing to tell the truth about how bad her work was while she had stomach flu.
Mercedes Martinez and Ivelisse Vélez to a lesser extent, given its mostly vitriol from Martinez rather than a mutual flow of it. Still, in spite of screaming about needing no one immediately before and immediately afterwards, as well as intimating that Velez was bad for the SHINE promotion, she still came to Ivelisse's aid against Valkyrie, under the reasoning "Boricuas stick together."
Cheerleader Melissa and MsChif started out as bitter rivals but seem to have reached a kind of tolerance\truce. The vitriol mostly seems to be on Melissa's side, seeing how she started their feud and laughed at MsChif's efforts to keep wrestling after being driven into steel stages and broke up their team to pursue MsChif's individual titles, the two still did agree to tag for awhile in between their individual bouts.
William Regal and Paul Burchill drifted toward this direction after Paul Burchill revealed one of his ancestors was Blackbeard's deckhand and started dressing like an old time Caribbean pirate because of it.
It seems a little time apart can solve any issue between Melina Perez and Jillian Hall. But keep them together for too long and they will end up fighting.
Perfectly personified with the angle between Kane and Daniel Bryan, two guys who really can't stand each other, but still somehow manage to work as a cohesive team known as "Team Hell No". They're also former WWE Tag Team Champions, a fact that the two both argue about by declaring that "I AM THE TAG TEAM CHAMPIONS!" whenever they win a match.