Generally I'd say yes, tip the artist. There are some things I would consider that would make me decide not to however.

If the artist owns the shop and is the sole artist, or is doing the tat somewhere out of the shop then maybe not. In those cases establish a set price and the artist is probably most, if not all of that money.

If it's an artist in someone elses shop, or even the owner in his own shop where several artists work then I think a tip would be in order.
Like any industry, tip according to the level of service you receive. If it's a custom piece, coverup, overly elaborate, or if you are making any special requests then a bigger tip is probably in order.

And no, tips don't always have to be cash... my first tat came when I worked at a pet store. The artist owned his shop and had a large fish tank in it, but could never get everything set up right. I fixed his tank problems, he hooked me up with my first tattoo.