To rank issues in Scrum Board's Plan Mode, you need the Edit Issues and Schedule Issues permission.
To rank issues in Scrum Board's and Kanban Board's Work Mode, you only need the Schedule Issues permission.

Changing the ranks in the Work Mode without the Edit Issues permission will still affect the rank values in the issue's rank field.

This seems inconsistent as they all use the Rank field. Also, the documentation says you need only the Schedule Issues permission.