I looked at this in a little more depth. I created a Classification Definition that had two Classifications, Schedule and No Schedule. I was then able to use those classifications to exclude Doors from a Schedule Table (as expected, having done so many times in the past) and was also able to use these to prevent the ability to manually add a Property Set to an object or style.
But even when every Property Set referenced by a Schedule Tag was set to Schedule, that did not prevent me from being able to tag an object whose classification was set to No Schedule, and the tags were anchored to the objects. After tagging, I did notice that the object-based Property Set referenced by the tag was not applied to the object, and the visible tag property value, which was in the object-based Property Set, remained at the default value. Surprisingly, the style-based Property Sets that were set to Schedule were still added to the Door Style that had been classified as No Schedule.
So, you can keep your cased opening Doors out of your Door Schedule, but you cannot prevent someone from tagging the cased opening. If tagged, the tags will not display any object-based property values, as those will not be attached to the object.
David Koch
AutoCAD Architecture and Revit User
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