This recently starting happening to me. And I have screen protectors
Screen protectors will only go so far, but they do lessen the damage significantly. As Night Fox explained, the screen protectors themselves will eventually wear away too and then the damage will be caused to the screen instead.
Protectors aren't expensive, so definitely consider replacing them. And as a last effort consider using a cloth, like a glasses cleaner or hanky. Place the cloth over the top screen and then close the 3DS. That'll make sure the two sides don't touch when closed.
Furthermore don't put anything on top of the closed 3DS, as that can push the screens down into each other.
Can't you just remove the two sharp edges? Like cut them off or something?
You could perhaps file them down, since the lip is higher than the bottom screen by a few mm.
However, at the bottom the lip is the same height and part of the back/home/start buttons, and if you filed it down you'd have to be careful not to damage the buttons.
So you could file down the lips around the screen and not the buttons, but the lower lip would still scratch the top of the upper screen.
