My shul has some "strange" minhagim that I haven't seen elsewhere. They reserve reciting the prayer for the ill for Shabbat and "regular" weekdays but will specifically not recite it if a Yom Tov occurs on a weekday. When I asked the rabbi why they have this minhag, he said he didn't know but this is a shul minhag dating back many years, and he has no idea why.
This could be a unique minhag to our shul. But, I'm curious if any group of people , nusach or otherwise does likewise or if there may be a source for this minhag. Obviously, I'm not including those places that have a minhag not to say this prayer in the shul ever or if they only reserve it for regular weekdays. I'm curious as to why there might be a difference between saying it on Shabbat but specifically not on Yom Tov.