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."
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, up until the point when he could.
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.
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.
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.
Internet writer Scott Keith invented the term "Wacky Tag Partners Who Hate Each Other" to describe this phenomena.
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.