I finally saw this a couple weeks ago. For pure entertainment value, irrespective of the actual concepts of music and musical education, it's objectively a good movie and entertaining to watch. JK Simmons played the hell out the character and demonstrates his fine acting chops.
As a serious musician who teaches private music lessons, the movie has so many things wrong with it I don't even know where to start. For that reason, I really disliked the film, and every musician I know who has seen it, especially NYC-based musicians and Julliard grads dislike it for the same reason. It is grossly unrealistic and factually inaccurate. Yes, I've dealt with my share of jerks, but there is no place in the world of music academia (or performing for the most part) where people who throw things at people and hit them are revered, respected, or even tolerated. While Fletcher eventually loses his job, he would never have lasted a single semester after his first bout of violent BS in real life and no other reputable music program would have hired him. Further, playing to the point of physical injury DOES NOT make you better, and it can actually make you worse. No serious music teacher will ever encourage or allow their students to injure themselves while playing - They will teach and encourage better, more responsible methods to avoid the risk of injury.
The lasting message it leaves with me is that it represents an unrealistic and profoundly negative view of serious musicianship and higher education that is going to turn young musicians away because they think that's what it's really like.