Update 18 January 2024:

In backlogs I'm seeing different behaviour depending on the field added to the card:
Jira Software company-managed Scrum backlog:
None still appears if the Due Date field is empty and on the card

Jira Software company-managed Kanban backlog:

If the watchers of this issue identify other fields that still show "None" on a card please add a comment including (a) the field name or custom field type and (b) whether it is on the board or backlog.

I believe this issue is fixed for how cards appear on boards:

Jira Software company-managed Scrum board:

Jira Software company-managed Kanban board:

Original description:
Issue
Currently it's possible to add additional fields to card views in Agile. It would be nice if it were possible to hide the field instead of displaying "None" if there's no value.

Workaround

Currently there is no known workaround for this behavior. A workaround will be added here when available