What does a covenant that Christians are not under, and a paragraph admonishing believers not to intermarry or be united with unbelieving philosophers have to do with modern-day tattoos?
There are a lot of things that we believe are still sins which are forbidden primarily in the OT. Christians do have beliefs with respect to grace and not being under the law in the same way when it comes to things like what we can eat and some other things, but that doesn't invalidate everything in that part of the Bible for us. The part about the nonbelievers has to do with the fact that tattoos are part of pagan worship and practice and Christians should not be taking part in it. One thing I've noticed is that many tattoos have occult imagery in them. For example why would someone put a star (pentacle) on their bodies? Why would someone put a dragon (symbolic of Satan in Revelations) on their body? It's even worse if they try to put "Christian" symbols on their body. Putting Bible verses on you and then sweating all over them or getting dirty in the normal course of life is a pretty big desecreation of God's word. Would you put a religious symbol on your arm and then wipe yourself or do other things using that arm? Imagine someone putting a tramp stamp of a cross on themselves right above their donkey... would that be majorly disrespectful? I think so. Then there are the 'tribal' tattoos. Why in the world would someone want something referring to a tribal (pagan) society?
Also think about the mark of the beast, why would we want to mark ourselves with pagan symbols when that's the very type of thing that will be done under pressure in end times prophecy (they will not be able to buy or sell without it) and it will condemn people to hell?