Piercings you can remove at your house (if you interact with the mirror), tattoos are permanent, unless you use the debug tool ("travel sphere" in the key items), and interact with one of the mirrors in the middle (the mirrors are on the northern wall). To exit the debug tool you then use the central magic circle.
There are a couple of paths which force a tattoo on you (in one of Sonya's paths, she gives you a small heart tattoo on your chest, and in the cowgirl path you get a "brand" on your butt).
Removing Sonya's tattoo might mess with her path, however, so I don't have any way to remove that one in the game as of now (I can add it to the debug tool with a warning, maybe).
Viewing post in Caliross, The Shapeshifter's Legacy comments
Those two make sense, and I didn’t expect story related ones to be able to be removed. Those don’t sound that bad at all, to be honest. If it’s directly related to the story I could see how removing it doesn’t make much sense, however I definitely did not want to keep the one that just says fucktoy lol.
The regular ones you can at least handle with the debug tool, then, no problem. Tattoos are mostly cosmetic, some NPCs might comment about them, but that's about it, with the exception of Sonya's, since I use it as a "milestone" in her path.
I did add the option in the debug tool for the other two tattoos anyway, since we talked about it (although it might be a while before the change makes its way to the public version).
Well, the debug room isn't supposed to be part of the game, but it's there for anyone that wants to mess around with it, which is why the features aren't really advertised anywhere. There are also some testing options in it (usually for my own use), so people should be careful when looking around in there.