This is subjective. "Health" here might be taken to include mental health, in which case there could be many positive benefits.
That said, there are some objectively bad health outcomes that can come from getting tattoos [1, 2]. Not that they necessarily will.
[1] infections, autoimmune response, etc: https://onlinelibrary.wiley.com/doi/full/10.1111/jocd.14498
[2] lymphoma: https://www.medicalnewstoday.com/articles/tattoos-may-increa...