Does a professor need a student's permission to cite the student's unpublished essay in their own work? Would professors keep an undergrad student's essay to contribute to knowledge in a certain field? I mean keep for the purpose of citing and using it as a reference when conducting further research on the topic. 

Or are they designed to just reiterate information already known just to prove that the student understands the material to qualify to pass a class?

Consider a scenario where the prof is covering course topics throughout the semester and then tells the students to write an essay (the typical introductory paragraph/thesis, body, conclusion, etc.). For example, a student writes about a certain topic in the field and the arguments used to defend that topic over an opposing claim against the original thesis. The student may present arguments that can can be a breakthrough in the field as to why a certain position on a topic is the best choice.

This could happen where the student has a background outside the classroom that contributed to the "original" discovery.
I imagine that undergraduate essays rarely contribute to the knowledge in a field, but there is nothing stopping them.

There is no rule that you have to hold a PhD to contribute.  You can be a graduate student, an undergraduate, or completely unaffiliated with a university and contribute.