The Fashionable Sunglasses is an object in the Existence, Now catalog. It is the 4th and final object sent to Sugar Plumps, which she wears on and off for the rest of her appearances in the game. Also, this item is used for the End Combo.
The sunglasses, while flammable, do not interact with the flames in any way.
If the sunglasses are dragged directly over the strange face in the back of the fireplace and let go there, it will become unfocused and fix itself to the face. However, it can still be interacted with by other objects, meaning it can be knocked off. If it is, it will refocus. If it is broken or burned, it cannot be fixated to the face anymore.