It seems like almost everyone I know has a tattoo and I even wanted one in high school myself. I have seen a number of things online about dangers of tattoos so I thought they should be posted here as well and let the parents decide what they think.
The main thing I read about face tattoos that was a turnoff (aside from obviously limiting job prospects) was that the ink can get into the veins that lead directly to the brain.
Other issues include infection, permanence, and the fact that while the FDA regulates the ink, they do not regulate the actual parlors so special attention needs to be paid to the tattoo parlor you choose.
If you are trying to convince your child to not get a tattoo, you may want to check out the second link below. It discusses some stats with tattoos that can be a turn off. Including that studies show that 1/3 of those that get a tattoo end up regretting it later in life and that the pain in getting them removed can be VERY intense.