Could getting a piercing cause allergic contact dermatitis?

Yes: One can develop contact dermatitis to the metal(s) used in the piece( ring, etc); the most common one is nickel. If one uses a topical antibacterial cream after the piercing is done to prevent infection( such as neosporin), the cream itself can cause an allergic dermatitis in some individuals.

Absolutely: 95% of the time it is a reaction to nickel in the "cheaper" jewelry that causes the allergic contact dermatitis. Stainless steel, silver, gold, platinum and titanium cause very few reactions. The cheaper the jewelry the more nickel present.