Geoff Tate and Queensryche in recent years.
Don Dokken and the band Dokken, Things got weird when Don wanted to form a solo act, also called “Dokken.” Guitarist George Lynch, who co-owned the name “Dokken,” took Don to court to prevent him from using that name. And the judge sided with Lynch; Don Dokken was not allowed to use his last name for the band. He decided on “Don Dokke,” and Lynch sued him again to prevent him from using THAT name. He lost that one.
Aerosmith and Steven Tyler.
I think most of the band - if not all of them - hated Tyler and may well still do.
After all, what is not to hate?
I saw him in a documentary one time where he said some really mean things about “fat people”. You know, the people who bought most of his shit.
David Byrne vs. the rest of Talking Heads (but Tina Weymouth in particular).
Kiss - I got Gene Simmons’ book a while back, and he was very critical of Ace Frehley, Peter Criss, Vinnie Vincent, and a couple other band members whose names escape me at the moment. Ace and Peter definitely got the worst of it though.
I don’t think he said anything negative about Paul Stanley or Bruce Kulick, but pretty much everyone else they’ve played with got blasted.
Quiet Riot - I don’t remember anyone ever having anything good to say about Kevin Dubrow.
I don’t know how well they get along at present, but Jack Bruce toured with Ringo Starr and his All-Starr Band in 1997. When they reached Red Rocks Amphitheater in Colorado, Ginger Baker, who lived in the area, joined them on stage, playing a few Cream classics alongside Bruce and Peter Frampton.
Yeah, but apparently everybody liked Topper. It was losing him as a personality buffer that seems to have precipitated the general dissolution.
Sadly Joe Strummer appears to have been one of those walking contradictions - an intensely charismatic, instantly likeable person who was nonetheless frequently a huge, selfish prick.
Ever seen Cass Elliott’s daughter? She never identified her father, but I’m 99% certain it was Denny.
John Phillips wanted her to have an abortion, and she refused because she wanted the baby.
I got Ace Frehley’s autobiography from the library a while back, and gave up about 50 or so pages into it because I just plain old did not like the guy.
As for Kevin Eyebrow :p, he was believed to have been dead for about a week before anyone noticed he wasn’t around. 
I’ve heard that Tom Scholz has had some rather unpleasant dealings with some of the lesser-known members of Boston over the years.
“So long, Frank Lloyd Wright” because Art G. majored in Architecture. FWIW I’ve always liked it.
If I remember correctly*, throughout the sessions for the “Bridge Over…” album they both were well aware they’d soon be parting ways.
Also, Paul singing “So long, Artie” in the fadeout might have tipped off Garfunkel about plans to “dump” him.
Not to mention that Mick attempted to kill Dave on stage in Cardiff.
The Smiths – Johnny Marr and Morrissey seem to loathe one another, at least on the latter’s part.
This sounds like the plotline of a 3rd rate daytime soap opera.
I just knew of the Byrne V Weymouth and her husband fallout(I forget his name). Maybe just because im a fan of Byrne I thought Weymouth and hubby were a bit up themselves. I think at one stage Talking Head albums were released with Tom Tom Cub songs on them.
Genesis started out as a bunch of musicians, and when Peter Gabriel started developing a rather prominent stage persona, the other band members really didn’t like it. By the time they recorded Lamb Lies Down on Broadway, Peter was on one room writing the lyrics while the other band members were in a completely different room writing the music. To this day, Tony Banks has nothing good to say about that album (apart from his own keyboard solos of course). Phil Collins and Peter Gabriel somehow managed to end up friends after all was said and done.
Styx was torn apart by Dennis DeYoung trying to take the band in one direction (Mr. Roboto, basically) and Tommy Shaw trying to keep it more hard rock and guitar based. Dennis and Tommy had a very public feud going on for years after Styx broke up. They kinda sorta buried the hatchet later, only to end up each going their separate ways again.
As was already mentioned, Paul McCartney and John Lennon used to get pissed off at each other quite a bit, especially towards the end. Each accused the other of ruining their songs. Ironically, their tension and different styles ended up combining to create really good music. That didn’t stop them both from being pissed off though.
Eddie and Alex Van Halen seem to have a lot of trouble getting along with people. Eddie in particular used to get really pissed off at David Lee Roth’s constant goofing around, even though Diamond Dave’s stage presence was probably responsible for a lot of their popularity. Eddie also got pissed off at Michael Anthony to the point where even though Michael got credit on the last album he supposedly played with the band, Eddie was so pissed at him that Eddie recorded the bass parts and didn’t let Michael anywhere near the project.
The Police, or at least Sting and Stuart Copeland, do not get along well together when they are (were) working. I think Stuart Copeland was quoted after the breakup as saying something along the lines that they discovered that they could all be quite good friends as long as no one mentioned music.
Frank Black and Kim Deal of the Pixies hate each other, and only reunited for tours because of really big financial incentives. Last I heard, Kim had finally left for good and had been replaced (and the replacement was then replaced).
Kiss has already been mentioned. I saw an interview where Gene Simmons basically said they were trying to be a successful band, not friends. I also wanted to mention that Paul Stanley has had many very bad things to say about Gene’s TV show and thinks that it was not a good thing for the band at all. The last I heard the two are no longer on speaking terms because of the show.
Was there something with the Eagles? I remember a Denis Leary bit from No Cure For Cancer, talking about Don Henley:
“I got two words for ya, Don… Joe Fuckin’ Walsh. How long’s your ponytail now, Don?”
No context given and the audience howls. I have searched for this conflict but never found anything.
I remember an article about 15 years ago about the touring antics of whatever’s left of Lynyrd Skynyrd. One member cut another member’s throat late at night on the tour bus when the victim was asleep. I’m sure they’ve worked it out by now, unless the guy he cut is the one who died a couple years later. Hell is it with us Southerners anyway?
That was Leon Wilkeson whose throat was cut. Ed King supposedly found him and his girlfriend Rhonda in a pool of blood. They were rushed to the hospital and stitched up and King was blamed for cutting Wilkeson’s throat. It was probably Rhonda who did it, considering they had one of those wild relationships (at one point Wilkeson had done three months in prison for beating up his girlfriend) and she slashed Wilkenson’s wrist a year later so badly that he could never play bass properly again.
Not quite a band but in the 19th century Gilbert and Sullivan at times had a strained relationship.
There is also John Fogerty versus the rest of CCR and Diana Ross (excuse me, Miss Ross) against Mary Wilson.