If this sounds like a dumb question to you, then my suspicions about myself are true…
Right, or wrong, I have been under the impression that every cell in the human body replaces itself every few years, using its supplied genetic code. So how come when tattooed skin cells die and regenerate, the tatoo doesn’t just erase itself in time. The particular information that constitutes the tattoo is not part of the genetic code… ?
IANATattooArtist, but I’ll throw out a WAG. 
Tattooing is the process of injecting pigment under the skin by puncturing it and injecting. It does not rely on destroying skin cells, though some tattoo forms do raise scars.
Another WAG, but I suggest that tattoos are actually scar tissue. Make sense?
I agree with Zagadka [puzzling, isn’t it ]. The only thing I have to add is that most cells have life spans measured in days, not years. Only neural cells (e.g., brain cells) last for years – for brain cells, a lifetime (they are not replacable…yet).
I can state with absolute confidence that DNA doesn’t enter into it.
I’m no tatoo expert, but I’d guess the pigment is injected into a layer of skin deep enough that it stays around for a while.
If the tattoo were merely in your epidermis, you likely would shed it over time. Tattoos go deeper than that, though, down into the dermis, which is tough and fibrous. Ink trapped in the dermis remains there a long, long time.
-prav, heavily tattooed doper
Most of the body cells replace themselves. The cells in your brain and spinal cord, however, never reproduce themselves. Of the cells that do reproduce themselves, they have differing lifespans. Some last for a few days, some for months, etc.
A tattoo doesn’t dye the actual skin cells, it injects large molecules of ink between the skin cells. Your body is unable to break down those molecules, so the tattoo stays for a long time. That’s how a plastic surgeon can use a laser to break up the molecules so your body can slowly carry them away, getting rid of the tattoo.
Zing.
“Tattoos consist of phagocytosed submicrometer ink particles trapped in the lysosomes of phagocytic dermal cells, mostly fibroblasts, macrophages, and mast cells” (Regarding Tattoos, Archives of Dermatology). That is, the tattoo is made of lumps of inert pigment, too big to be carried away by the lymph system, that have been ‘swallowed’ by cells. When those cells die, others come along and do the same.
[slight hijack]
Can we see your tats pravnik?
I’ve a large tiger done in purple and black with bright blue eyes on the left side of my chest. Measures probably abour 6" by 5".
[/ slight hijack]
Looking up more tat information, I’ve also come across the explanation found by aeropl and raygirvan.
What raygirvan said, in technical terms is:
Ink gets trapped in skin. Ink particles are to big to be “sweated” out and too small to be attacked by your immune system. Thus, permanent ink.
IANA Tattoo artist, but I have’em and have spent hours upon hours hanging out with my piercer and TA friends in their parlors.
broccoli!
too small to be attacked by your immune system.
Not exactly. Immune system cells can attack the ink particles, and get do get as far as swallowing them - they just can’t digest them. Size isn’t the problem; they can’t break them down chemically, so the stuff just stays walled off inside the cell.
I don’t know about the OP, but I just wanted to second raygirvan’s point. The immune system attacks viruses, and I’m pretty sure they’re smaller than globs of ink.