I have heard rumours to the effect that Colin Baker and Sylvester McCoy went through a somewhat strained period, but that probably had rather a lot to do with the way Baker Secundus was removed from the show.
As for companions: it is part of Fan Lore that nobody liked Matthew Waterhouse (Adric), and Lalla Ward (Romana II) was known to be "difficult". The companions of Doctor V (Peter Davison) didn't see eye-to-eye much either, but that might be because they were bored - they were badly served by the scripts.
Anthony Ainley (Master II) was a luvvie's luvvie, and apparently very intolerant of horseplay on the set, and hit it off badly with Tom Baker from the start (I think there was history there, too).
C Baker I think was very critical of Christopher Eccleston's decision to leave the show after one series - understandably so whatever Eccleston's reasons were (reading between the lines I suspect that the pressures of the first of the new serieses and everyone getting used to producing a show on that scale made it hard on the cast and especially the crew - and Eccleston felt that the crew were being badly treated as well as finding the pressure and publicity rounds involved with the role difficult to handle - I think by the time David Tennant came in a lot of these problems had been ironed out and Tennant a more extroverted character with limitless energy was more suited the demands of the role - but this is just my conjecture) it must be galling to see someone leave the show after such a short period of time when you wanted to stay on for ten years and got unceremoniously booted off before you were ready to go.
Anthony Ainley may have been a lovey but apparently according to the DW Livejournal thread I was on for a bit was well liked by fans whom he treated well - one lady even had a massive crush on him. I got the impression that for some reason Pertwee just thought that Baker wasn't good enough for DW right from the start - both of them I think had diva tendancies which didn't help.