Tattooing on the tongue.
Yes, it is possible. No, the pigment won't hold.
In the 1820's the Hawaiian Queen Kamamalu received a tongue tattoo to show her grief after her mother in law's death. This was witnessed by a missionary named William Ellis who commented that the queen must be in great pain, she replied; "He eha nui no, he nui roa ra ku‘u aroha." (Great pain indeed, greater is my affection.)
In fact the pigment will hold if tattooed with a syringe. By using an injection method of tattooing the tongue a “forced blow out” can be created under the skin. This is done by injecting the pigment into the "permanent layer" of the skin, as it’s commonly referred to in tattooing. Careful consideration has to be made not to inject the pigment intramuscularly.
Using a tattoo machine to tattoo the tongue doesn’t work very well because of thick layer of taste buds lining the top of the tongue. The bottom of the tongue can be tattooed with a tattoo machine and will hold because it lacks the thick layer of taste buds that the top does.