No, not that I've noticed. I use cocoa powder for brown in my soaps frequently. The only thing it can do is rub off on scrubbies or wash cloths but I've never had it stain. I've tired micas and like the color from cocoa powder better. Sometimes I'll add a touch of black to make it darker if needed.
I have never had it fade in about 4 years. I still have some soap colored with cocoa powder that is as dark as dark chocolate and I made it at least 2 or 3 years ago.