This feature is there in Superchess but only applicable to so-called walls: dead non-moving pieces occupying one or more squares. The owner was allowed to annihilate it ("opheffen")
Suggest a name for this chess variant!
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It looks like this variant was invented back in 1971 or earlier. From The Classified Encyclopedia Of Chess Variants by D.B. Pritchard and second edition edited by John Beasley (cool book by the way, can be downloaded at https://www.jsbeasley.co.uk/encyc/encyc.pdf):
In Suicide Chess [piece removal]
(origins unknown, and not to be confused with
Losing Chess), a player may remove one of
his own men from the board instead of
moving. (Feenschach, May 1971, also
Berloquin, 100 Jeux de Table)
Sep 10, 2019
0
#4
Hahah, thanks: I wanted to take a look at that book for a while, but couldn't find a link. Well, looks like I've been beaten by some solid 40+ years then! 
I'm still surprised that this variant has not gotten any attention though: it seems very simple to play and is subtly different from regular chess without any gimmicks like weird pieces or some completely new game concepts…
Greetings all!
I've been fooling around with a chess variant that I invented some years ago (have never seen it mentioned elsewhere) and it seems quite interesting, offering unexpected attacking opportunities and changing dynamics of the endgames quite a bit. The only change is this new rule:
• A player may remove one of their own pieces (except the king) from the board, as long as it doesn't put their own king in check.
Clearly, in most normal positions one would not want to spend a move on removing their own piece from the game, however, there definitely exist circumstances when doing that may help your position quite a lot. Some stalemates are also not possible, as one has to "sacrifice" all of their pieces before running out of legal moves.
Can anyone suggest a catchy and unclaimed name for this?
Thanks!