This is what's called a "teachable moment" - there's a difference between doing the right thing and doing the wrong thing/not getting caught. If your son isn't driving yet, he will be soon, so it's not unlike speeding if there's no cop around or running a stop sign if there is no other driver. You do the right thing.
Now, is the rule unreasonable? The way you've explained it, yes. The thing to do is to have your son write a well-reasoned letter about the policy. This could be to the principal or to the librarian (with a copy to the principal), or to any committee that oversees the library (see if there is a media committee or something similar). Finding out who is in charge and who is authorized to make changes is part of your son's investigation. This is an intellectual exercise.
These resources are in the school but not being fully utilized. While there may or may not be justification for keeping certain materials reserved for specific students, do they restrict library books as well saying that only students in certain classes can check them out? What is the justification for keeping materials in the library on weekends when someone else could be taking advantage of them? Your son could bolster his argument with an approximate count of the DVDs in the library and the likely price (conservatively estimated), then multiply it out. Is this a $10,000 inventory collecting dust? If these DVDs are worth having because of their educational value, why not make them available to the entire student body? There could be a deadline as someone else suggested - after 4:30 on Fridays, they are available to anyone, and due back at 8 AM on Monday. Students who don't return them could be fined just as with any library book.
Your job is to help him craft this letter, and to tell him you will not condone his stealing. Your reaction "to be careful about this" really condones the "do it and don't get caught" philosophy which I think is really going to hurt your credibility as a parent. Instead, teach him about proper civil discourse and how we go about changing rules (and laws) in this country.
This has nothing to do with whether he "needs" the DVDs. It's about a school resource that is rationed, and since education is supposed to be the great equalizer, it's baffling why a school would prohibit responsible students from taking advantage of expensive materials. But it's up to your son to argue that point.
If he wants to turn this into a petition with other students signing on, that's the American and democratic way.
My son and his friends were opposed to a school policy that suddenly began restricting cross country training routes. A teen had been hit by a car (freak accident with neither the teen nor the driver paying attention), and then the athletic department banned training runs on a bunch of other streets (not the one where the accident occurred). So a group of students put together a statement which analyzed the routes and acknowledged the need for safety but questioned the efficacy of this policy. They even had tee shirts made up listing the forbidden streets and created a campaign to "free" these streets. They found out how to get on the School Committee Meeting agenda, they waited their turn respectfully, and they presented their argument. They showed respect to the administration by stating that they understood the concern for safety, and by suggesting some safety precautions that could be implemented (running in a tight group, wearing reflective gear, etc.). It was outstanding. They didn't win out, but they gained the respect of the principal, the School Board, the Superintendent and Assistant Superintendent. They learned how the democratic process works by participating in it. But they didn't just flaunt the rule.
Teach your son that anything worth fighting for is worth doing well. Meantime you will help develop his ability to respectfully protest, his ability to research and write a cogent response to this policy, and help him negotiate with the powers that be. This will be terribly strengthening to him regardless of the outcome.