But first, let's go over what parts of the film ARE definitely classic tragedy. Right of the bat we have Neil Perry, a good old fashioned hero. He was born into a noble bloodline; anyone attending the Welton academy has parents who can afford to send their sons to boarding school. He has a tragic flaw in the way he thinks the only way he can live is by acting and not having the strength to resist his father's wishes. And (most importantly) he suffers a huge downfall, he kills himself on a cold winters night. Was it the will of the Gods? Hubris? Fate? I'm personally on the fate side. As a "romantic" Neil was destined to die a death in the name of love (of acting).
This movie also runs parallel to Greek Tragedy in a few other ways. The film contains several "odes" in the form of Mr. Keating's impassioned lessons to his pupils. Mr. Keating acts as the Choragos in this tragedy by being a leader to the students (who are the chorus). In the end of the film when the students stand to say farewell to their beloved teacher it almost seems like a classical exodus chanted by the chorus.
Now, I contend that The Dead Poet's Society is not a true tragedy for one major reason, the characterization of the chorus. In Greek tragedy the chorus is a mildly unsettling group of people in masks who walk across the stage while chanting odes and poetry; the individual members have no strong or distinct personality of their own. In TDPS however, every boy in the chorus has their own unique character and I would contend that some of them are tragic heroes in their own right. Take Charlie Dalton for instance. He is of the same noble bloodline as the other students, has the tragic flaw of hubris, makes major mistakes in publishing an article in the paper and punching another student, undergoes a major change by assuming the name "Nuwanda", and ultimately has the downfall of being expelled. In a classic Greek tragedy, a member of the chorus would never get such a complete character ark. A true Greek tragedy would never have such development of minor characters! In fact, at times it seemed like our "tragic hero" Neil Perry wasn't even the film's main focus.
On a side note, I'm glad that this film breaks the tragic mold. I think that Greek tragedy is a really kind of restrictive form. I mean, having more than two characters onstage at once was considered crazy innovative in those times! And don't even get me started on those masks... (shudder)