Looking at Muppet Wiki's "what links here" for a number of images, it seems like when a character does a new version of their signature song, the old version doesn't really air again. I'm mainly referring to when it's just the original character doing a new rendition, not if a celebrity does their own version (whether the original character is with them), or if they have completely different characters singing it (can't think of many such instances, though).
Looking at "what links here", so far the wiki notes only two episodes past 136 to include the original Rubber Duckie (last week when video footage was found for episode 1505, it turned out that that was one of these two to have the original - the guide previously listed the more-famous season two version, so maybe the scripts don't include a difference in the various versions, maybe there's more listed that the wiki thinks is the season two version), that's gone on to become the most iconic version of the song. Season 30 included the famous season 2 version in two episodes, as well as a film insert that had the recording play over it, but then later that season they introduced another new version with Ernie, and the season 2 version hasn't aired since (though the season 30 version hasn't aired in too many episodes).
C is for Cookie is basically the same deal, with the original being shown for so many years before it was remade in season 30 and the original was never broadcast again. In fact, I can't remember the last time I actually saw the original on the show (I know it's been in plenty of late-1990s episodes, but the last time I personally remember watching it on the show as opposed to video/online was sometime before 1993).
I don't think the original ABC-DEF-GHI or I Love Trash were repeated on the show, but there were quite a few versions. I feel like the late-1980s ABC-DEF-GHI and the season 22 I Love Trash were the only ones repeated as inserts. Maybe some of the early 1970s versions were repeated as inserts and I'm just not sure (I want to say that the early-1970s version that sometimes gets snippets shown in clip shows and such was repeated).
And Bein' Green is an interesting case. Looking at "what links here", the wiki currently doesn't list it in any episodes past 286. It probably did air between seasons 3 and 7 (maybe later) and the wiki just doesn't note it yet, but the fact remains that there's no known airings of that one since they remade it with just Kermit in season 7. The original never aired in the 1990s, and I think the season 7 version only had one '90s airing. Seems like the Lena Horne version was the go-to version to air on the show.
And yet in most cases, when these segments have been released on video, the original ones/most iconic ones have often been released the most while other versions have had very few video releases. The true original Rubber Duckie hasn't been released on video but the more iconic one has been released several times (Sesame Workshop tends to think it's the original so there's that), even after introducing the season 30 one (the only home video release combines it with the film version, though I know that the region 4 Playtime with Ernie has it, if that counts). Even after the season 30 C is for Cookie was introduced, the original is often released (including twice this year), while the remake has only been on video twice (one of which was as part of a mash-up of the many different versions of the song). The original I Love Trash, which wasn't really an insert, has actually been released on video as often as the season 22 version (and we did get an early 1970s version on video around the time that the season 22 version came out). The original ABC-DEF-GHI (which also wasn't really an insert) has only been released once while the iconic late-1980s version has been on video at least three times (and will be on the 50th anniversary set).
It seems to be more the case with the signature songs of main Muppet characters than the human cast. While The People in Your Neighborhood tends to be best associated with Bob (though many of the cast has had their turns with the song), most "cast songs" tend to more commonly have new performances, and I think the previous versions have often been shown after a new one. Big Bird and Oscar have had many versions of their signature songs, some just being one-time street scene performances, but they also tend to be part of "the cast" more than the Muppets (if it makes sense).
It's also interesting because for a long time, there didn't seem to be restrictions/limits to showing existing material with older versions of the puppets. I don't really know why they needed to remake Rubber Duckie or Bein' Green. As they were still showing material from the early years in season 30, it's a little odd that they remade songs like Rubber Duckie and C is for Cookie with the original characters/original vocal tracks, but that season's curriculum was to re-familiarize fans with classic songs so it makes some sense when you consider that.
I've been thinking about adding a lot of this info to TV Tropes' "first installment wins" but I'm not entirely sure if it should count there.