Well, lets check the list.
The downfall of the hero/heroine: Neil, our hero, commits suicide. He realises that this is his fate, as his parents are not accepting of who he is, an actor, and are forcing him to give uo that part of him and become a doctor, as they want.
Hero/heroine encounters limits: Neil's limits are his father. He lies to him about the play. But his father is constantly dictating what Neil must do, his classes, extra curricular. His father acts as a fence surrounding Neil's passions, and once Neil hops the fence he is trespassing and metaphorically shot on sight.
Includes a tragic hero(es):Neil is our tragic hero.
Bloodline- He attends a prestigious school
Flaw- he is cowardly and doesnt stand up for what he believes in and and what he wants to do with his life
Mistake- lying to both his father and his teacher
Change- he decides he is going to do what he wants and he follows his passions
Downfall- he realizes he cant be who he wants to be so he commits suicide
Anagnorisis: The final scene where Keating, the teacher, returns for belongings. There seemed to be an underlying sense of fate and destiny that I felt the characters understand,
Viewer feels Catharsis: Sure, personally not for my, but i know plenty of people who when they watch this movie are always purging their emotions,
Odes: The meetings of the society served as the odes, as they occurred between the scenes of actions. Each one was focused on a different poet.
Choragos: Keating is the choragos, he started the meetings but leaving the book, and all his notes are written in the book and used at the meetings initiating them. Even though he never actually attends the meetings, he is essentially starting it.
Exodus:The final scene where the students stand on the desks chanting "Oh captain my captain" This shows the significance keating has made on their lives and the fact that they wont be the same again.
SO yes, It is a modern Greek tragedy.