No. Only creatures are subject to effects that require saving throws.
Spells that do damage based on a saving throw explicitly restrict their effects to creatures, including cantrips. Two examples of many:
A vertical column of divine fire roars down from the heavens in a location you specify. Each creature in a 10-foot-radius, 40-foot-high cylinder centered on a point within range must make a Dexterity saving throw. A creature takes 4d6 fire damage and 4d6 radiant damage on a failed save, or half as much damage on a successful one.
... Each creature in a 15-foot cone must make a Dexterity saving throw. A creature takes 3d6 fire damage on a failed save, or half as much damage on a successful one.
Ioun stones aren't creatures, so none of these spells (or any other spell that targets creatures) can affect them.
Some such spells also have specific effects against objects; two examples, again of many:
.. ignites flammable objects in the area that aren't being worn or carried.
A nonmagical object that isn’t being worn or carried also takes the damage if it’s in the spell's area.
Neither of these would affect an ioun stone; when in use, it's treated as being worn or carried, and when not in use, it's neither flammable nor nonmagical.