In response to the above answer:

Being a tattoo artist does not give someone the legal capability to randomly dab someone's skin in blotter fashion hoping (or not in this case) for a reaction.

There are SEVERAL chemical compounds that make up tattoo ink. One of which is arsenic, which MANY people are allergic to and WILL cause adverse effects.

If you’ve difficulties swallowing or breathing, it's an allergic reaction (and not the kind you treat with benedryl), the physicians can give you an epinephrine pen and a hand full of antibiotics wishing for the best.

worst case scenario, you hope your tattoo artist used newer inks that can be removed by process of injecting lithium into the effected skin.

Answer:
You can be allergic to the ink, but how severe it is (whether it makes your throat swell shut, or give you a rash) really depends. If you're honestly worried about it, ask your artist to give you a dot test or a patch test.. i forget what it's called, but they put a small amount of ink somewhere on you thats not too visible to see how you react.