In the OT it speaks in Leviticus about cutting for the dead and not printing marks next to a verse which seems to say about not trimming/marring beards. Most Christians would say that they are not bound to imitate OT Jews with bushy beards; being a New Testament believer under grace, pursing Gospel testimony opportunities is different drastically from being OT Jews in the land under the law. But if someone does indeed have a conscience against getting a tattoo, then this should be respected indeed also.
In the Bible, with the complete New Testament, saving faith in the Lord Jesus and His work at the Cross for sinners is a real key concept: John 3.16; Romans 5.1; Hebrews 11.1-12.2, etc.