Quest items don't have weight. Drop all your stuff and you'll see.
Quest items don't weigh anything, no, but there seems to be some bug in the game that prevents you from dropping/selling the same NON-quest item. For example, The amulet of Talos is at some point a quest item, but along my travels I found two other Talos amulets, and I couldn't sell them, because one out of the now three ones was a quest item. Maybe they corrected this in some patch (haven't patched my game).