Unread posts by instructors
In the years I've been using Moodle it has not been possible to select who you'd like forum notifications from - it's all or nothing from a forum. There is development to enable subscription to one thread within a forum - see MDL-1626.
Remember however that the News forum is a place where teachers can post announcements. These are read only for students and everyone has forced subscription.
You could show the students how to change their subscription options in their profile to Digest which then then provides a daily list of all posts to the forums they are subscribed to. This list displays who each post was from so they can take note of those from the instructor.