There's actually an even bigger problem here.
Within production, work is usually subdivided by rank. Some tasks are normally allocated to the senior person, some to the middle people, some to the apprentices/assistants/juniors. These ranks aren't uniform across companies (and the exact tasks can vary widely from position to position), but within companies themselves, "we usually have the apprentices do that" is a common refrain.
In many companies (especially in non-commercial) there is now a class of work below that of the apprentice: the intern.
The catch is that, while the existing ranks provide for an obvious progression (a few seasons of apprenticing will qualify you for a mid-range job, which--in a few years--will tee you up for a senior position), intern work often doesn't. A season spent making photocopies and serving canapes to donors and picking up dry cleaning and fixing the artistic director's Blackberry in no way prepares you to move up in the world. (Networking is great and all, but should you need to do fifty hours a week of busy-work in order to make whatever contacts you can make at some summerstock company out in the boonies?) Unless there's a clear path between your internship and the job you eventually want to hold, you are wasting your time. Worst-case scenario (and several people have had this experience), you gain excellent qualifications to hold unpaid internships, but no qualifications to find paying work in the field.
At the other extreme is the overqualified intern: the person who is absolutely vital to the show. If you're performing duties at your unpaid internship for which someone else would normally be paid, you're eating your own lunch. (Do you really expect to be paid in future for performing work which they can easily get someone to do for free?)