Teaching a MOOC with Coursera has made me aware of all the anonymous posts in the system. Coursera allows students to post anonymously on a per post basis. There is a checkbox that allows a forum post to be anonymous. I wonder about the value of this. I can't imagine an anonymous student in one of my physical classrooms.
I think, within a classroom, you should be responsible for what you say. This is a place that can teach you to be serious and considerate. All classes, beyond teaching the syllabus should teach you to be considerate, empathetic, and respectful. A class should teach you to consider what you say before you say it. Posting anonymously allows you to be inconsiderate, un-empathetic and even rude.
Allowing anonymous posts allows students to make "off the record" comments, which in a large way defeat the usefulness of a permanent record of a student's involvement in a class.
Is there a place where anonymous is valuable in a classroom?
Anonymous posts are great for critiquing the teacher in a traditional school. In that role they work beautifully. A student, when critiquing a professor, needs to know that they will not be punished for their comments about the teacher. I have received some of the best student to teacher critique in the anonymous surveys in my traditional and online classes. I thank my students for taking the time to let me and the administration know how my teaching has been.