Just before coming attractions the movie theater houses typically have a set of short movies telling the audience how to behave.
I'm not looking for "Coming Attractions" or "Our feature presentation" videos. The types of videos I'm referring to for ask their customers to not talk, turn off their phones, and don't make a mess.
What is the term for such clips?
In my cinema we refer to them as Etiquette Cues, and I have heard others refer to them as Policy Cards (when it is a written text), but I do not believe the industry has a consistent terminology for them.
More broadly, any additional material in-between adverts/trailers is referred to as an insert, or a message displayed on the screen as the audience enters (which often conveys just such a message) is a holding slide.
It is called a Policy Trailer. It outlines the theater's policy on smoking, cell phones, fire safety, and of course the concession stand. Large theater chains produce their own trailers. Smaller theaters use one supplied by Coca-cola or Pepsi. Here is a link to a long list of them: http://cinematreasures.org/video/categories/policy-trailers
The general term for these that I am aware of is PSA's or Public Service Announcements.
While the site referenced by @darth locke is not movie specific, it does address the general concept of what a PSA is, and the purpose they serve.
I have also come across the terms "Pre Curtain announcement" or "Pre Show announcement", however these are more related to theater. Given that movies often used to have curtains (And some still do have motorized curtains), it is possible that it could be applied to cinematic applications.
