Even if a task does not use fields from the Shared Data Template, the user must still have the appropriate permissions for the entity type involved in the journey. If the entity is an Entity Group, which means the user must have permissions to update an Entity Group in order to complete any data task applied to its draft.
The permission required is:
EntityGroupManagementEdit
Users who do not have this permission will not be able to complete the data task, even if the specific fields used in the task are not tied to the Shared Data Template.
Other tasks in the journey may still work if they do not require updates to the Entity Group draft, which is why the user can complete some tasks but not the one involving data updates.