How can I use Freire's thesis in the banking concept of education to analyze the student-teacher relationships in the film Dead Poets Society?

Freire's thesis is indeed a revolutionary one. To begin, it would be a good idea to briefly distinguish between the "banking" concept of education and what Freire calls "problem-posing" education.
The "banking" concept of education centers on the transference of information from one party to another. In school, the teacher transfers or "deposits" information into the minds of students. This type of education is sterile, fact-focused, and orthodox in nature. Absolute rules govern the teacher-student interaction, leaving little room for communication and true learning.
Meanwhile, "problem-posing" education focuses on exploration, creativity, and critical thinking. Now, we can discuss the Dead Poets Society movie. This film stars Robin Williams in one of his most iconic roles: he is Mr. John Keating, an English teacher at Welton Boys' Academy. If we refer to Freire's treatise, we can analyze the movie based on two main elements:
1) The banking concept of education creates a mythological concept of reality. This reality is what teachers impart to students. It defines reality according to a set parameter of values. Additionally, it is authoritarian in nature and brooks no dissent. Meanwhile, problem-posing education demythologizes the conventionally sterile concept of reality.
Let us take Mr. Keating's first class. He comes in and begins whistling a medley of tunes. This is unusual behavior for a teacher. The students do not quite know how to react. Meanwhile, Mr. Keating circles the classroom and eventually walks out. Now, the students are flabbergasted. They have yet to greet Mr. Keating in the manner they have been taught. Now, their English teacher has just unceremoniously left the classroom.
In only a few minutes, Mr. Keating has already begun to demolish the conventions of education. Next, he beckons his students to follow him. Now, I am going to cut to the chase. Mr. Keating effectively tells his new students to "seize the day." What he means is this: if learning is to occur, the students themselves must put forth the effort to reach for knowledge (instead of merely imbibing it from the teacher's mouth). From Mr. Keating's perspective, the teacher and his students are partners in learning. Students are not mere spectators, they are also "creators." This is in line with Freire's problem-posing educational prototype.
2) The banking concept of education resists dialog. Meanwhile, problem-posing education encourages dialog as a form of learning. Essentially, it welcomes dialectical learning (dialog between two parties that may hold opposing views), rather than didactic learning (where one party teaches the other).
In the movie, Mr. Keating has a student read an introduction to poetry by one J. Evans Pritchard. The short reading focuses on the conventional way of analyzing and appreciating poetry. When the student finishes reading, Mr. Keating orders everyone to tear up the introductory chapter. He scathingly refers to the introduction as "excrement."
Initially, the students are bewildered and shocked by Mr. Keating's order. Here, I must acknowledge that many educators find Mr. Keating's attitude and approach to literary analysis troubling. Certainly, there is a place for the mechanics of poetry analysis. Mr. Keating's main point, however, is also worth noting:

This is a battle. A war. And the casualties could be your hearts and souls. . . . Now in my class you will learn to think for yourselves again. You will learn to savor words and language. No matter what anybody tells you, words and ideas can change the world.

Critical thinking is more than a blind adherence to rules; it is also about exploration, creativity, the testing of hypotheses, and cross-examination. None of this can occur under the conventional "banking" form of education. From Freire's treatise:

Projecting an absolute ignorance onto others, a characteristic of the ideology of oppression, negates education and knowledge as processes of inquiry.
Indeed, the interests of the oppressors lie in "changing the consciousness of the oppressed, not the situation which oppresses them"; for the more the oppressed can be led to adapt to that situation, the more easily they can be dominated.

The banking concept of education is about conforming and control, rather than "true reflection" and "creative transformation." In your essay, you may also decide to compare the teaching methods of Mr. Keating to that of Mr. McAllister (who teaches Latin). In the movie, note Mr. Mcallister's reaction when he catches the students ripping up J. Evans Pritchard's introduction to poetry.
Hopefully, my answer will get you started on your own ideas about analyzing the movie according to Freire's treatise!
http://puente2014.pbworks.com/w/file/fetch/87465079/freire_banking_concept.pdf