I've heard that it helps your tactical skills, but can a chess variant really help you in tactics more than simply playing standard chess.
Bughouse really doesn't. What can help you is shogi. Shogi is similar to chess but with complications that start almost immediately and become much more crazy than in chess. The reports that I have had from multiple players who went from chess, to shogi, back to chess is that the chess tactics suddenly seem exceedingly easy when you return.
To speculate for a moment, bughouse is harmful because it is similar enough to chess that it interferes with your thinking about chess. Shogi is sufficiently different to be separable.
I initially felt bughouse (or crazyhouse) would definitely help improve one's chess. I thought this since you have to do the extra thinking of where you would optimally place the captured pieces that you have. But when I tried playing it, its way different and difficult and went out of hand pretty quickly for me. So I feel there's hardly any use trying to play bughouse inorder to improve your regular chess, as it is a completely different game altogether.