Consider this dramatization of Apollo 13 as an explanation of why it is probable we don't know what the most heated argument has ever been in space:
Astronauts worry about how ground control's perception of their performance will affect their opportunities to fly again. Space agencies worry about how the public's perception of their performance will affect their budgets.
In Packing for Mars, in chapter 2 author Mary Roach quotes Norbert Kraft (who now is the chief medical officer of Mars One) talking about events during his participation in an isolation experiment to evaluate factors in a crew's behaviour on a long deep space mission. The experiment, by the Russian Institute of Medical and Bioogical Problems, lasted 110 days:
...when Russian men bloody each other's noses, it's a 'friendly
fight'. (Kraft confirmed this surprising item. "It's how they settle
disputes. They did it on Mir.")
Now, he didn't elaborate (or didn't give Roach permission to reveal more), and that's hearsay only. However, in the same part of the book, Roach talks about multiple sources reporting vodka and cognac were very commonly smuggled aboard Russian missions, and the first part of the above quote was a comment by Valery Gushin, a psychologist with the Institute that did the isolation study.
The crew of Soyuz T-14 returned to Earth from Salyut 7 several months early because the mission's commander, Vladimir Vasyutin, was ill. There have been varying reports that the illness was prostate infection, and depression, but this British Interplanetary Society article seems to present the full picture from the crew itself:
All of a sudden, the look on Volodya Vasyutin’s face changed and he
stopped talking... The irritation that this conversation
caused caught my attention. Before bedtime he floated over to me and
started talking about his health... On 16 October Savinykh wrote he was
concerned about what he described as Vasyutin’s ‘old ailment’, which
Vasyutin confided to him had begun after underwater EVA training in
the hydrotank at Star City. However, they decided not to consult with
Mission Control before the spacewalk and press ahead with EVA
preparations. Savinykh also noticed signs of depression and
irritation: ‘The music irritates him. I’m curing him with medicine and
advice. The best medicine is work.’ Another two weeks later Vasyutin’s
condition had deteriorated to the point that the cosmonauts had no
choice but to report the situation to Mission Control.
The salient thing here is that the crew only reported this serious illness after they felt they had no choice. So, this seems to suggest there are many things they don't report at all. Ground control finally elected to terminate the mission three weeks later after all further attempts to treat the condition failed - and it isn't like the crew hadn't already tried using the medications on board. One of those attempts included bringing in a psychic. There has been a lot of airing of opinions that the main issue was really depression, not infection. Roscosmos was not at all pleased and considered witholding the traditional awards given to cosmonauts who fly missions.
The crew of Skylab 4 staged a one-day 'strike' in which they refused to work on the grounds they were exhausted because ground control had been pushing them too hard. The three rookie astronauts on that crew - Jerry Carr, Edward Gibson, and Bill Pogue, never flew again. Pogue and Gibson left NASA within 2 years of the mission. This may have been for personal reasons and not to do with any possible impact on their career caused by their little mutiny. Still, it is the kind of thing other prospective astronauts might well see as a cause-effect relationship.
Based on the above, I assert that probably we don't know what the most serious interpersonal conflict in space has been, because what happens in space, stays in space.