Someone suggested this would lead to lots of draw by insufficient material. I really don't see why that is. It's not like players will just be taking their own pieces willy-nilly until they have none left. Taking your own pieces will typically hurt your position more than it helps.
Smothered mates are eliminated, but at least you're winning material if you force the opponent to capture his own piece.
Not just smothered mates are eliminated, but it becomes harder to trap a piece. For example, sometimes you can trap the opponent's queen because some escape routes are blocked by their own pieces. In this variation, the queen could escape by taking a less valuable piece or pawn. Still an advantage to the attacking player, just less of one.
You may want to exempt the king altogether -- no taking your own king and/or the king can't take his own pieces.
Initially it sounds interesting, but it seems this would do nothing more then make it harder to win(no smother mates)
Again - smother mates are pretty rare in games anyway.