According to https://asana.com/guide/help/email/email-from-asana you can get a notification if a task assigned to you has its due date changed. If the task is not assigned to you, you won’t.
I can see two solutions:
#1 Educate users. That would be the best obviously but the hardest.
#2 Develop a code using the API to watch for due date changes on certain tasks and notify whoever needs to be notified (maybe the manager given in a custom fields on this task).
However there is a third solution apparently . According to https://asana.com/guide/help/tasks/dependencies if a task you depend on has its due date changed, you get on notification. So you could have a master task, depending on several other tasks, and you would get notifications if one them has its due date changed. Voilà!