I have a problem with Briar Hearts being flagged as quest items, despite no current quest requiring them. Is there a console command to remove the quest status or is it permanently bugged?

Are you doing or have you done the quest to fix the White Phial? A briarheart is required for that.

I completed it, and the subsequent quest, and they are still flagged, it is pissing me off quite severely.

Eat that stack of briar hearts. New ones that you pick up later then won't be quest items anymore.

Lytha is correct. If you happen to have a briar heart BEFORE you finish the white phial it will continue to mark ALL of your subsequent briar hearts as quest items as well (if you keep stacking them). The only thing you really can do is either eat all of them or use up all of the ones in your inventory for potions.

They don't appear in the list to make potions, would using a console command to drop them work? or do they have to be removed from the game? (removeitem or eating)

If they don't come up for potions just eat all of them.

Do they have a low carb option?