I'm going to revive https://secure.phabricator.com/T11557 (which was merged into https://secure.phabricator.com/T4863 which did not really implement what's my understanding of the feature's scope) here.
https://secure.phabricator.com/T7930 also mentions Due Dates.

Current Situation

It's possible to configure this via a custom date field in your instance, e.g. going to /config/edit/maniphest.custom-field-definitions/ and defining

{
	"deadline.due": {
		"name": "Due Date",
		"view": true,
		"edit": true,
		"description": "Deadline for completing the task.",
		"search": true,
		"fulltext": true,
		"type": "date",
		"copy": false
	}
}

Current Limitations

We've done this in our instance, however it's flawed as there's no sufficient visibility:

• Due dates are not rendered in task result lists or in task cards on project workboards
• Overdue tasks assigned to a user are not rendered as some item in the top bar for that user or via some other notification
• Overdue tasks are not explicitly listed on project pages with <blink><marquee>HTML markup</marquee></blink> as a popup banner blocking the entire screen

I'm not saying that all these should happen. But something to allow folks to realize that you're hitting a Due Date.

For comparison: GitLab

For comparison, this is how GitLab renders a Due Date in the task list and the side bar of a single task:

See also: Maniphest Points

This feels a bit similar to the case of /config/edit/maniphest.points/ which had a similar history and ultimately ended up being codified via /src/applications/maniphest/constants/ManiphestTaskPoints.php, /src/applications/transactions/editfield/PhabricatorPointsEditField.php, etc.

Out of Scope

Potential separate followup tasks (as scope creep can easily make tasks non-actionable):

• Trigger some automated notifications (maybe to achieve via https://we.phorge.it/rP19be32656f84b43ede8e6a5bb84a2e3c8ac7ac40)
• Integration into the prototype Calendar application