I suppose it depends on the intention of the tattoo.
Christ had nothing to say on the matter. Paul did comment re 'temple of God' but was not referring to tattoos.
In Middle ages and after 'Christian' colonists associated tattoos with heathen practice (another excuse to kill or enslave people and steal their land - but that was unrelated to Christianity other than in name)
I have met clergymen with tattoos.
Churches, too, are presumably temples of God, too They are usually painted, and often have beautiful stained glass windows, and art (sometimes the art is outside as well as inside).
I was under the impression that God judges you by what you are, and not by your appearance. But I suppose there are Christians who will say Christ was wrong there, too!
In some areas (not Western), tattoos are an important part of their cultural heritage - yet they are Christian. I never met any Christian clergy that are there, who object, or try to interfere with the practice, other than to try and assist with hygienic methods of applying them.
Some Christian Cultist/extremists may object, but such can hardly be called real Christians, as in followers of Christ and his message, anyway.