This is obscure enough to matter only to the obsessive type so I never bothered to pose the question. I also was unable to find a solution with a search for Pin or Unpin.
Ever pin an object and wish you had not? But even if you unpin it, ever wish you could eliminate the icon? Ever want to remove the pin from a family you are modifying without leaving the unpinned icon in place? Removing pin entirely instead of unpinning it only? We found the way to do this. There is no logical reason this works but I have tested this on walls, grids and reference planes. Makes a cleaner file.
Unpin the object or reference plane or whatever. Draw two other objects or reference planes on either side in any non-equal location. Create a dimension string for the three objects and set the equality constraint, thus moving the formerly pinned object. Select the formerly pinned object and you will see the pin/unpin icon has disappeared. You can use geometry already in place and can choose to delete the dimension or not depending on your goal. The key is the equality constraint.
Like I said, pretty obscure. I sometimes wonder if these things are intentional parts of the program that actually matter for something important (and hidden from the teeming masses) or if these things are just an entertaining treasure hunt.