Blazing Guy festival is a parody of Burning Man festival. At one point, one of the participants mentions Burning Man instead of Blazing Guy.
When Mr. Lassen punches through the wall between the fourth and second grades, Milhouse puts his nose in the hole and says "Here's Milie!" That is a reference to a famous The Shining scene "Here's Johnny!" This scene was already referenced in Treehouse of Horror V segment "The Shinning".
When Bart and Milhouse are thinking of a way to humiliate Lassen, Milhouse mentions a website link called www.nzfilmhere.nz. Originally if you happen to type this link (or click on it right now) It will take you to the Fox Simpsons website with a picture of Milhouse saying "Made Ya Look!". Now it redirects to https://www.fox.com/the-simpsons/
Milhouse: (Peering through a hole in the wall) Here's...Millie! This is a reference to a famous line in the movie The Shining, which is itself the trademark introduction of Johnny Carson by Ed McMahon during The Johnny Carson Show.
Mr. Lassen cut Bart's hair, however it is ABH (actual bodily harm) to cut someone's hair without permission.