As you can probably imagine, this answer will lack precision, because we do not know for sure what high school teachers typically ask. Moreover, I am sure that there will be quite a bit of diversity. In light of this, let me suggest some possible essay topics that I feel are popular, at least in the North East.
First, with any comedy, a popular essay question asks you to consider the not so comic elements or even tragic elements of the work. This tension goes back as far as Plato's work, Symposium. For example, Beatrice and Benedick insult each other often, and Claudio and Hero do much of the same (with great humor), but what are the tragic elements?
Second, another perennial favorite theme and possible essay topic is the comparison between men and women. How are they portrayed and what are the standards for each of them. Is there a double standard? For example, does fidelity and faithfulness function the same way in men and women? All you need to do to answer this question is to examine Benedick, Leonato, Claudio and Don Pedro and compare them to the female characters.