For the satirical undertones card your film must include satire, which is the use of irony, exaggeration, or ridicule to expose the faults of a person or group. Often satire is applied to society in general, and social commentary is one of its most common uses. A film does not need to have a deeper social meaning to be satirical, however. It simply needs to aim to mock its subject, usually by taking it too seriously or too far.
A good example of satire is The Truman Show, in which the main character's entire world is a television show, with him as the unwitting star. The plot revolves around Truman's discovery that his world is crafted for him by other people, his identification of the few genuine relationships in his life, and his eventual (spoilers) escape from the show into the real world. The movie satirizes a number of things about American culture, including the power of the media and our seemingly unquenchable thirst to know the intimate details of others' personal lives. In the film, the producer of the TV show (Christof) literally takes on the role of God in Truman's life. In the outside world, some people leave Truman's show running 24 hours/day in order not to miss a moment.
Harry Potter and the Order of the Phoenix is seen by some as a satirical look at government interference in education, which the story definitely takes to the extreme. It suggests that involving the government too much is a surefire way to make sure kids' schooling is dominated by political agendas, not useful learning that will help them in real life. I disagree with the view of this film as satire, because the main characters in the film take an open stand against this philosophy. The theme is certainly discussed, but it is done in a straightforward manner, not the backhanded slap in the face that satire generally relies on.