I am interested in this as well.
I believe there are problems with unpredictable feedback from LTI providers which makes a due date (as used in quizzes and assignments) difficult to implement. However, being able to set a due date so that External Tool activities appear on the calendar would be very helpful.
Being able to set "Allow submissions from" and "Cut-off date" dates as well would be nice.
I have added an issue in the tracker about this:
Please vote for it if it's something you'd like to see.