Do you think students should do a teacher observation? Why or why not?
I am assuming that your question concerns student observations of teachers for the purpose of a course in education, meant to give a student at least a glimpse of what it is like to become a teacher. I cannot think of any reasons why students would not want to do this, or why teachers would not welcome this, and I can think of many reasons this is a good idea.
For a student who is thinking of becoming a teacher, this is a valuable experience, to see how teachers function in the day to day environment of the classroom, from a different perspective. We all have had years and years of observing teachers from our vantage points as students, but observing a teacher from the perspective of one who is interested in becoming a teacher is very different. We notice a great deal more this way, how the teacher has planned, how the teacher manages the students, and how the teacher handles content. This gives a potential teacher a deeper understanding of what it is really like in the classroom, from the other side of the desk. For some students, this is an experience that dissuades them from becoming teachers, which is good, since it is better to know sooner rather than later that this is not what they want to do. For most students, this is a good way to learn something about teaching, before the onset of practice teaching. Many teacher education programs require one or more teacher observations for this purpose.
Cooperating teachers, I have found, are happy to take time to answer questions, to share lesson plans, and to discuss what they do every day. They like to encourage people who are interested in entering the teaching profession, and they can offer valuable advice. Sometimes the teacher who is observed will be a cooperating teacher for practice teaching, and it's nice when a relationship with a student teacher has already been formed.
I think back quite fondly to the teacher observations I did as a student, one of whom became my cooperating teacher for practice teaching. I knew already I was in good hands. This is a great starting point for anyone who is thinking of teaching, and I can't think of any reason a student or teacher would be against this.