Timothy Dalrymple added some insights into the reasons beneath various types of tattoos. I have to underscore that the whole list is worth reading, but the last one is especially interesting:
4. The Flesh Made Word. More common in general is the practice of tattooing Chinese characters, and more common in Christian circles is the practice of tattooing Bible verses or biblical or theological phrases. This is especially interesting in the light of the theology of the LOGOS and the incarnation. In the incarnation, the LOGOS, the eternal Word, became flesh. The LOGOS transcended the world and its changefulness, representing the eternal truth and the power by which all things were called into Creation. But when a Christian tattoos a Bible verse or a faith-phrase upon her body, she makes her body into a text. She reverses the incarnation of Christ; in her de-incarnation she is making the body, what is prone to messiness and effluvia and decay, into a true and eternal Word. They are turning themselves into the Bible, or a part thereof.
I really appreciate Tim's point here. However, I am not sure the person who tattoos a verse has dematerialized their own body any more than the person who gets the Harry Potter tattoo. The text is just as much a part of Magritte's classic as the pipe, and last I checked no one is tattooing themselves in Times New Roman. In other words, the decontextualized verse or stock Christian cliche has an intelligibility that is unique precisely because of its location on the body (or its use on the canvas).
There's a question here about how "immaterial" the Bible actually is. It may be a different sort of materiality than the body, but it is material in its nature through and through. The symbols of the Bible are as much a part of the material world as any piece of art, as the illuminated manuscript tradition of the Middle Ages was only too happy to point out.
None of this is antithetical to Tim's point. If anything, I'm trying to point out that it needs to be broadened in two directions. First, Tim's point about the dematerializing effect of linguistic tattoos seems to apply to tattoos of any sort. Somewhat paradoxically, the decision to inscribe the body points to the body's insufficiency on its own, outside the remarking that we do to it. Because of that, those who tattoo themselves with religious symbols are fundamentally no different than those who use the decontextualized verses or stock Christian cliches.