I'd venture to suggest that 'The Three Doctors' is worse than in its immediate predecessor, except that I might get lynched; certainly, I find 'The Time Monster' easier to sit through. I was going to start by describing what I like about this story, but I couldn't really think of anything. So I'll start with the least negative thing I can think of.
The interaction between the different incarnations of the Doctor is of course the main selling point of 'The Three Doctors'. It is unfortunate that William Hartnell was in such poor health by this time that he has a very minor role, appearing only on the TARDIS scanner. I'm certainly not churlish enough to criticize Hartnell for being terminally ill, and I'm glad that he got the opportunity to play a role in a story celebrating the anniversary of the series the success of which he was instrumental in establishing. Whilst Hartnell's health means that the First Doctor is a mere shadow of his former self, there are brief glimpses of the indomitable old character from his era, and this is probably my favourite aspect of 'The Three Doctors'.
The other two Doctors however, are rather less entertaining than they should be. Troughton seems incapable of putting in a bad performance, so I suspect the problems with his performance here stem from the script, which reduce his Doctor to a caricature. Try and recall the Doctor who appeared in 'The Evil of the Daleks', 'The Tomb of the Cybermen', 'The Seeds of Death', or 'The War Games' and compare him with the Second Doctor here; the Second Doctor's fierce intelligence and palpable sense of urgency during times of danger are played down here so that his clown-like characteristics can be brought to the fore. Possibly Baker and Martin feel the need to pigeonhole him into the description ascribed to him by the First Doctor in their script, but the result is a Second Doctor who is in most ways watered down, and in one or two others grossly exaggerated. His constant prattling about his recorder annoys me just as much as it annoys Omega.
Unfortunately, Pertwee isn't much better here either. As in 'The Time Monster', he seems to be on autopilot, and for the second story in a row this results in a threat to the entire universe being woefully underplayed. A good example is during the Third Doctor's scenes with Omega before his previous incarnation turns up; presumably, the Doctor is both humouring Omega and also suffering slightly from the fact that he is overawed at meeting a legend. Sadly, Pertwee fails to convey this and instead the Doctor just gives the impression that he's wondering when Omega will get around to offering him a cup of tea. As with 'The Time Monster', compare this with the Doctor's sense of urgency in 'The Dæmons', which despite being another story of which I'm none to fond, does demonstrate how well Pertwee can portray urgency and a sense of impending doom. The actual interaction between the Doctors has some merit, but personally I only find it interesting when they are cooperating; the bickering between the Second and Third Doctors seems to give Troughton and Pertwee their cue to start hamming their parts up.
So that's the Doctor's out of the way; let's move onto the villain. Omega is potentially a rather sympathetic villain, who is clearly shown by the script to have been driven insane by millennia of solitude. In practice however, he's an over-the-top pantomime villain, thanks to Stephen Thorne's painfully unsubtle performance. Especially cringe-worthy is his bellowing cry of "Whaaaaaaat?!" when he discovers that the Doctors have escaped. When Omega is in jovial mood, Thorne sounds as though he's reading his lines for the first time and has had time to practice. When he's angry, he just sounds melodramatic without actually being convincing. I noticed on this occasion in particular just how stagy the scenes set in Omega's palace actually are, due partly to the sets (the rather tacky walls contrast unpleasantly with a bare studio floor), and the fact that Thorne gives the impression throughout that he is concentrating very hard on spotting his next cue.
Then we have the Time Lords. There are two ways to demystify the all-powerful Time Lords first seen in 'The War Games'. The first is to present them as a thoroughly corrupt race of politicians as Robert Holmes does later in 'The Deadly Assassin'. The second is to have them portrayed by wooden actors spouting stilted dialogue in a day-glow nightmare of a set. Roy Purcell is especially dire as the President. And there's very little else to say about them.
What of the other regulars? Katy Manning is as good as ever, but Jo is present purely so that the presence of two Doctors can be explained in very laboured fashion. She does very little else here. She also plays a role in another cringe-worthy moment, after the Doctors have conjured up a door in the cell in Omega's palace. Again demonstrating the complete lack of anything approaching dramatic tension in this story, she and Benton are the last to leave after the Doctors and Tyler. Rather than rushing quickly out of their prison, they laugh about the others having all the fun and jog casually through the door. Nice to know they aren't worried about the prospect of universal Armageddon then
Benton is generally OK here, and I'm extremely grateful that with all the rubbish present here we are spared the addition of Mike Yates. However, Benton gets another terrible moment, in the TARDIS in episode one; the Second Doctor appears, the two Doctors start arguing and trying to explain the situation to Jo. Benton remains silent for a couple of minutes and then suddenly exclaims "Doctor! Where did you spring from?", as he greets the Second Doctor. This is again an example of bad scripting and sloppy direction, as it rather makes it appear that he has been waiting for his cue and has missed it because he hasn't been paying attention. And then there's the Brigadier
By 'The Three Doctors' the Brigadier's character has reached an all time low. The intelligent military leader of Season Seven is long gone and has been replaced by an imbecile. The exaggerated coziness of the "UNIT family" has become so absurd here that Lethbridge-Stewart is reduced to a comic foil. Benton, one of the Brigadier's subordinates, has no problem understanding the events taking place around him, accepting two Doctors and the transportation of UNIT headquarters with ease. The Brigadier on the other hand understands nothing, presumably because Baker and Martin mistakenly think that this makes for a witty time for the viewer. In short, the Brigadier seems far less intelligent than his sergeant, which cheapens his character enormously. He's also lost the considerable air of authority that he once had, now seeming petulant rather than commanding when he orders Benton to search for Tyler for example.
I've mentioned this briefly already, but the script is diabolical. It is almost inconceivable that a story a mere four episodes long and featuring three Doctors could feel padded, and yet this is the case here. In episode four, as the various humans captured by Omega return home, we get a very tedious few minutes as they step one by one into the column of smoke. They could have been transported by a wave of Omega's hand, or stepped into the column as a group, but instead we get unnecessary prevarication in order to pad out the episode. In episode one, the explanation of how there are suddenly two Doctors present is both laboured and strangely over-complicated. This is not a complex issue at its most basic level, and based on the model of time travel adopted by the series; if you could travel in time, you could meet your younger self. Explaining regeneration is perhaps more difficult, but the Brigadier seemed to cope with it well enough in 'Spearhead from Space'. Here, we get a very tiresome and drawn-out interchange as the Doctors try and explain things to Jo, and therefore the viewers. It feels extremely patronizing and contributes to the feeling that 'The Three Doctors' is padded. Episode Two is even more padded, achieving little except to offer a cliffhanger very similar to that of Episode One, but on a slightly larger scale. And speaking of cliffhangers, the fight scene at the end of Episode Three is one of the worst in the series to date.
There are only two supporting characters of note; Ollis and Tyler. Ollis is utterly superfluous beyond the opening scenes of episode one, suggesting that Baker and Martin suddenly realized that they were stuck with him and decided to have him tag along with the Brigadier and the others. Tyler is present simply to ask scientific questions to the Doctors, which results in the Doctor explaining things to the audience through Tyler rather than Jo. The entire story is a self-indulgent mess and a poor excuse for a celebration.