I didn't ever heard of a similar variant before.
It's interesting: pieces that move along "lines" (Queens, Rooks and Bishops) can become very powerful, since a player can remove his own Pawns to increase the range of his pieces. Imagine a battery of two Rooks and the pair of Bishops looking towards a castled King: they could really be a powerhouse, since the attacking player can volountarly "open lines" himself.
On the other hand, all the other pieces are a little bit of crap: who cares about the "jumping ability" of the Knight when anyone can open a blocked position by sacrificing a Pawn? And the King will find himself in trouble, when defending: no more "blocking behind the enemy Pawn", since the "enemy Pawn" can easily disappear the next move.
I think that this variant would lead to open and very aggressive games: both players can sacrifice material at will in order to open lines.