Sometimes the only difference between an unpaid intern and an employee is the pay. If the internship is purely a training experience, the intern doesn't work for you and therefore can't be fired, fairly or otherwise. If you use an intern as free labor, on the other hand, she may have the rights of an employee, including the right not to be unfairly terminated.
Employee or Intern
It doesn't matter that you call someone an intern -- labor law focuses on what he does, not the label you stick on him. Several tests distinguish an intern from an employee. To provide a true internship, you should be providing an educational environment where the intern actually learns new skills. The intern must know going in that the work is unpaid, and his activities must be for his own benefit, not your company's. The intern mustn't displace or take work from regular employees, and his activities should be closely supervised -- for example, you would typically assign an employee to oversee an intern. If you pass all the tests, you have an intern; if you flunk, you have an employee.
Wrongful Termination
If you use your "interns" as free employees, you could be in legal hot water. In 2013, interns at several companies sued their employers, charging that they weren't educated and were used as free labor. An intern claiming wage theft could sue you for at least minimum wage pay for the hours she worked. If you dismiss an intern before the internship is up, that can also cause trouble if a court rules she's an employee. If you fire her for a legally invalid reason -- punishing her for making a sexual harassment complaint, for instance -- she could sue for damages.