A due date is set on checklists either from the template, if the template has a due date or can be set on each checklists manually.
When a checklist is near the due date, all assigned people in the checklist will be notified by email. If a checklist passes the due date assigned people will also be notified that the checklist is over due.
To add, edit or remove a checklist due date open the checklist page and click on Set due date in the action menu.
This will open a flyout with a calendar where you select the date the checklist is due. Click the Add due date button to add the due date to the checklist.
The due date is now visible in the header on the checklist page.