I just had occasion to revisit a star on the PEP program -- IS Gem -- which had been declared non-variable, and I noticed that it was still being observed, and is still non-variable. The same is true of CK Ori, which was a subject of a forum post. And a list of L-type pulsating red giants published by Percy & Terziev in JAAVSO 39, 1 (2011). So how can we "turn off" the observations? Janet Mattei used to say that the stars might become variable (again) in the future, but most if not all of them were never variable in the first place.
People need to be aware that these stars do not vary, but they can serve a purpose as "standards" on which different observers can compare their results. Limited sampling of these stars could be very useful for revealing systematic problems in data collection and reduction.
As long as we identify them as likely non-variables in VSX I would opt to leave them active to allow us to use them to compare instruments and observers. One example of this is alf Com (actually alf Com is likely an eclipsing binary, with a very long period). Both Tom and I have observed this star recently which has allowed us to intercompare our systems. Our results are within a few millimags of each other despite using different technologies (solid state diode vs photomultiplier tube). Although we could set up a separate list of 'standards' the current database ensures retention of the observations for the future, and easy access with no additonal work on our part and little downside.
"Life is short - eat dessert first!" - Unknown
If AAVSO observations were the only source of variable star photometry in the world, there might be a slim case for collecting observations of constant stars. The AAVSO observer advantage is, in my opinion, the deployment of interested observers on targets worthy of further observation. There is no case to be made to continue to encourage and collect observations of manifestly constant stars.
This is partly an issue of leadership and guiding and shepherding talent and effort in the direction where it will have the most impact in the current era. My opinion is that we should stop enabling the collection of observations into the AID when stars are not variable.
People are free to continue to observe constant stars. But we don't have any obligation to encourage that choice. Science is progressive.
The value of the constant stars is for identifying problems in observational technique and data reduction. No one is proposing observing these stars as a significant fraction of effort, but they would be very useful for spotting trouble, trouble that is likely widespread and going undetected at present.
Attached is a light curve of one of the constant stars referred to at the beginning of the thread. Identities of the star and the observers have been omitted. Even excluding the biggest outliers, the magnitude range is 0.2 in V band.