Gorillaz: Particularly in Phase 1, 2D never seemed to harbor any resentment towards Murdoc, who ran him over, put him in a coma and fractured his eyeball, ran him over again and fractured his other eyeball, knocked out his two front teeth at some point, regularly beats and insults him, and stole several of his possessions, at least one of his girlfriends, and (according to the website) most of his internal organs. 2D still once announced, "Aww, I love you, Murdoc!" in an interview, much to Murdoc's annoyance. He attempted to stand up for himself when the band re-formed, but evidently it hasn't gone very well, as he's currently being held captive by Murdoc on Plastic Beach. He did recently declare that he hates Murdoc, so he seems to be coming to his senses, either because of time spent away from Murdoc or because he's reduced his medication intake.
The Police. Despite catapulting each other to stardom, frontman Sting and founder Stewart Copeland could not agree on anything, even coming to blows during the recording of their last album, and their clashing egos severely alienated guitarist Andy Summers. However, after permanently disbanding in 1986, Sting invited the other two to his wedding in 1992 and they performed an impromptu gig for the other guests. The three came together one last time in 2007-08 to play 152 shows on a worldwide tour: on opening night, Copeland wrote a disparaging review of their performance, which finished by describing how they fell into each other's arms laughing backstage.
Steven Tyler and Joe Perry. They were actually called "The Toxic Twins" back in the day, though this may have had just as much to do with their incredible drug habits.
John Lennon always dished out his fair share of constructive criticism and potshots at Paul McCartney in the media post-Beatles breakup, but he truly loved and admired Paul and said he would not want to hear fans or journalists slag him off, thinking that that was what John wanted to hear. John very often defended his friend, and he greatly praised Band On The Run when it came out. Likewise, Paul greatly admired John, and downhearted stopped sessions for his Tug Of War album for several months after learning of John's murder (he wrote "Here Today" in tribute).
Tommy Scott and Jamie Murphy of Space. Franny Griffiths as well - one interviewer described him and Jamie as practically married to each other, other interviews stated he had a Big Brother Instinct towards Jamie, and in an interview with Q he said of Jamie, "He's a difficult little fucker. But it's not the same without him. We all buzz off him."
Tommy and Franny are this in the current line-up, as the two oldest members and the two remaining original members. They take the piss out of each other constantly onstage (and often in interviews as well), but they're clearly friends off it.
Noel and Liam Gallagher of Oasis. Noel once said that they love each other, they just cannot stand being near each other.
Rod Stewart and Elton John seem to have this kind of friendship. They exchange lighthearted swipes at each other, play practical jokes on each other, sometimes fall out at times or have major disagreements, but are truly the best of friends.
Joey and Johnny Ramone had this relationship to a T. Johnny would make frequent anti-Semitic jokes at Joey's expense, who would in turn refuse to play any songs of Johnny's with any political content. When Johnny started dating (and later married) Joey's then-girlfriend, Joey fired back by writing "The KKK took my baby away", and got Johnny to record it before he understood the context. However, the two collaborated closely on everything, and when Johnny received word of Joey's death, he slid into a deep depression which stuck with him for pretty much the rest of his life.
In "I Haven't Seen You In Forever" by They Might Be Giants, the narrator and his old friend are at first glad to see each other, but the two part ways after being reminded of how much they annoy each other. Ultimately, the two start to miss each other again and make amends.