FWIW, Roseanne Barr’s eldest daughter Becky was played by two different actresses (one of whom is now on Scrubs) on Roseanne, and Elizabeth Montgomery’s husband Darrin was played by two different actors on Bewitched.
They even had the second Becky say she preferred the second Darrin in her first episode of Roseanne. Nice. 
Nicer still was when they did the morphing faces intro on Roseanne, the first Becky’s face morphed into the second, then back to the first when that actress returned. Sadly by that point in the show, that was usually the comedic high point of the episode.
Swapping actors out isn’t a continuity issue, because the new actor is stepping into the existing role with the character’s backstory intact. For example, Becky when played by Sarah Chalke was still the oldest child, still married to Mark, still a high school graduate working to support her husband through trade school, etc. just as when she was Becky when played by Lecy Gorenson. Same character, same continuity, just a different actress.
The idea of two actresses playing Becky actually became a running gag on the series: one episode ended with a Donna Reed parody about the two Beckies ("Meet Lecy, the one you used to see from '88 to '93…), and another episode had Becky’s first appearance prefaced with an announcement: “Ladies and gentlemen, the part of Becky, previously played by Lecy Goranson, then by Sarah Chalke, then again by Lecy Goranson, will be played tonight by Sarah Chalke.”
I have my videotape of the original airing. Nibbler’s shadow is there.
Monty Python’s Flying Circus.
That would be a parody of The Patty Duke Show.
That business actually started with the very first episode with Gorenson back in the role. Every time Becky walked in a room, someone asked “where the hell have you been?”
That’s what I meant. I always get those The First Name Last Name Show shows mixed up.
Technically he wasn’t really a different Harry Kim. OK, he was, but not all that different. The ship was duplicated and it was, what, a few hours before the “bad” one blew up and that Harry went to the “good” one along with the baby? So it’s not like his entire life he was different, because only up till a few hours ago, he WAS the same Harry Kim.
But Futurama is by no means perfect. As already said, Star Trek was mentioned before the episode “Where No Fan Has Gone Before,” with no ill consequences. Also, the percentage of metals and alloys making Bender’s body is well over 100%.
However, it should be known, that the TV show with the best continuity I have ever seen, is The Venture Brothers. In two seasons, I have only noticed one mistake, and that was due to the producers forgetting that one episode needed to be aired after another one, and so forgot to tell as that. Even then, the error is small (a minor character meets someone for the first time, but in an earlier ep, we see she has already met him.)
One of the worst continuity examples is the amazing disappearing character - the one who is never spoken of again. From Adam Cartwright to Yeoman Rand to Chuck Cunningham, they simply cease to have existed like George Bailey.
For crying out loud, even the Cunningham parents forgot they had an older son by the time the series ended.
Of course, there’s the Frasier Crane school of correcting continuity errors –
Martin: [about Frasier] Hey, Sam, what’d he tell you about me, the father, the old cop?
Sam Malone: Well, uh, he told me you were dead.
Martin: [surprised] Dead?
Frasier: Well, we had an argument one day. He called me a stuffed shirt and hung up on me. I was mad.
Sam Malone: [to Martin] You were a cop?
[to Frasier]
Sam Malone: You told me he was a research scientist.
[Martin reacts]
Frasier: [to Martin] You were dead! What did it matter?
I subscribe to the theory that each season of “MAS*H” exists in its own time/space continuum.
One sort-of continuity thing that irritated me about MAS*H was that there were two episodes that revolved around Potter encountering an old buddy of his who wasn’t fit for command. In one ep he was vehemently pissed about the idea of taking him off active command and in the other he led the charge to get him removed. If you’re gonna tell the exact same story twice then the outcome of the story should probably be pretty similar.
Except that you find out eventually that she’s a clone, so it’s different clones that are dying, not her over and over, so not really a Continuity error
In the animated series?
That’s certainly the premise of the movie (which I thought was pretty good), but not of the MTV animated series, IIRC.
All this talk about changing actors/actresses, and nobody points out that The Munsters had two completely different Marilyns (Pat Priest and Beverly Owen). (And the movie had yet a third – Debbie Watson)
Or the fact that the original Adventures of Superman had two different Lois Lanes – Noelle Neil and Phyllis Coates.
My take on the OP was different… I thought about it as meaning what shows are there where continuity wouldn’t be an issue. For me what came up were shows like “Sliders” and “Quantum Leap” and “Voyagers!” where the character(s) were in a completely unique situation each episode.
In “Voyagers!” the characters would travel in time to events in history and interact with the setting of the time. Then you’d never see any of those people again in future episodes. In “Sliders” the group moves to parallel universes each episode, where everything is completely different… since they don’t return to those universes in future episodes, there is no continuity issue to maintain–the writers are free to pretty much have whatever happen happen without consequence. In “Quantum Leap” it’s the same way… you almost never saw any of the characters again after the episode ended, so no continuity to maintain.
Days of Our Lives introduced not one but TWO young male characters with the intention of having them be gay. When both of the actors caught on with the female viewers, those story lines went right out the window. Which is how Eric Brady and Brady Black ended up straight.
Except they copped out in the movie. She was not cloned over and over. She did not die in the movie. In fact she was the only one who wasn’t cloned more than once. It made it the exact opposite of the TV show. It would have made for a much better movie if she had died
I’ll grant your premise, but “almost” carries a lot of weight in that sentence, Opal. We learn about the great loss in Al’s life very, very early in the first season, and it is mentioned repeatedly through the series, and winds up being the very last thing “which once went wrong” we see Sam set right. Then there are characters like Jimmy LaMatta and Donna Eleese who make return appearances.