We'll explain this using one of the most famous tattoos in pop culture: Mike Tyson's temple of flame in The Hangover. When The Hangover Part II homaged his appearance by having one of the main characters drunkenly get a similar face tattoo, it got Warner Bros. sued by Victor Whitmill, the artist responsible for the original. He alleged that by reproducing the tat and including it on posters and other marketing materials without his permission, Warner Bros. committed copyright infringement. The studio tried to claim the second tattoo was a parody, showing off how stupid the original was (making it symbolic of the whole Hangover franchise), but the judge disagreed and Warner was forced to settle out of court.
Warner Bros. Pictures
Their real punishment was having to admit involvement in The Hangover Part II.
It's an issue that's coming up more and more in video games as well. When the popular basketball game NBA 2K created its virtual avatars, it also added their many tattoos, leading to a group (or inkling) of tattoo artists suing the makers of the game for duplicating their artwork. Their lawsuit only failed because they filed for copyright after the game's release, not because it was dumb, but they still managed to claim damages for loss of income. In another case, when a tattoo on MMA fighter Carlos Condit appeared in UFC Undisputed, the artist responsible filed suit on the same grounds of copyright infringement.