Chhers John, but where on earth did you hear that? My first impression was that it is complete nonsense. On thinking it over, perhaps it is something peculiar to sensor-shift IS or even the Olympus implementation of it. Or possibly it's just a myth. I have certainly never heard of such a thing applying to optical IS (as used by both Nikon and Canon, and also by Tamron, Sigma and Tokina).
Further, I have successfully used IS with high shutter speed (up to 1/8000th) for many, many thousands of shots with a wide variety of lenses, including the Canon 70-300L, 100-400 Marks I and II, 500/4 Mark I and 600/4 II. Nowhere in the Canon literature is it recommended to turn off IS for high shutter speeds, and Canon are generally very good about providing that kind of advice. A Nikon person might be able to confirm my assumption that the same applies to Nikon's VR.
Canon has not released a lens which requires IS to be switched off when using a tripod since some time last century, and similar remarks apply to Nikon's VR lenses. I'm not sure when Sigma, Tamron and Tokina IS implementations became tripod-safe, but I rather suspect that they always were insofar as none of the third-party makers had an IS product at all until well after Canon (and probably Nikon too) had dealt with the tripod compatibility problem. It was only ever an issue with a handful of very early IS lenses, all of them long since discontinued, and few of them likely to be still in daily use after all these years.
Of my long-lens shots (these are the one where I am most likely to be using high shutter speeds), I handhold about a third, take another third or perhaps a quarter using the car windowsill as a rest, and the remainder with a tripod. I never turn the IS off except for flight shots, and often not even then. Here is what happens when you use a Canon IS lens with a tripod at 1/8000th. (In this case, it was a Canon 500/4.)
That's not a fluke shot. That sort of clarity is what one can expect as routine, given (a) nice light, and (b) no silly mistakes by the photographer.
(Why did I use 1/8000th and a high ISO? I was set up for flight shots and this little chap decided to sit still.)
Here is another one: 1/4000th, tripod, Canon 600/4 II.
Years ago, some people used to recommend switching IS off at higher shutter speeds, not for any direct image quality reason, but because IS systems used to slow down auto-focus. Not by much, but enough to make a difference. And it does (this is why I turn it off for flight shots), or rather, it used to. On my old 500/4 Mark I, for example, it can make a slight but visible difference. With newer IS lenses, AF in the preferred IS Mode 3 is lightning fast. (I should be careful here not to exaggerate: AF on the older IS L Series lenses is still very quick indeed. We are not talking chalk and cheese differences in AF speed with IS on or off, just a moderate effect which is usually worth tolerating in exchange for the extra sharpness of a stabilised image.)