Some bad things can happen as a result of tattoos (e.g., infection or hepatitis), but these can happen regardless of what is illustrated by the tattoo. In fact, tattoos of any sort are probably best avoided. See “What does the Bible say about tattoos?”
When the Bible talks about “demon possession”, it refers to illnesses that have no visible cause. It was the way people spoke about illnesses such as epilepsy, schizophrenia, muteness and deafness because they had no other explanation of these illnesses. The New Testament refers to these illnesses and disabilities in the same way that everyone else in the first century world did — as caused by invisible “demons”. That doesn’t mean the demons are real.
Tattoos cannot cause these sorts of illnesses.