Many doctors express concerns about getting new tattoos. The tattoo itself may not be unsafe, but the tattoo-getting process introduces some unnecessary risks that could easily wait until after the baby is delivered.
Most doctors will cringe at the thought of getting a tattoo while pregnant because of the risk of Hepatitis B. Hepatitis B and HIV are passed through unsanitary needles, and are very dangerous to pregnant mothers. It can be passed on to unborn babies with disastrous results.
There isn't enough research to tell us the effect of tattoo dyes and inks on unborn babies. Inks do contain chemicals, and since we don't know their impact, it's just easier to avoid them until after the baby.
Some hospitals and anesthesiologists have a policy against administering an epidural as pain relief to a laboring mother who has a tattoo on her lower back. It might be wise to check with your hospital to find out what their guidelines are if you plan to use an epidural.
The other consideration regarding getting a tattoo in your pregnant state, is body changes, that could alter the appearance of the tattoo, giving you unintended results. You could be bigger or smaller, tightened up in some places, stretched out in others. What started out as a butterfly could shrink into a housefly or expand into a pterodactyl. Color and line quality could be affected, as well.
Many tattoo artists won't service a pregnant woman for these reasons, and also because their artwork could easily be affected by the obvious body changes. They also would rather avoid the liability that comes with the potential risk of infection, and the weaker immune system of the pregnant customer.