Inbox Messages tab in the Business App can be turned off under the user-level settings in the Partner Center.
- In the Partner Center > Navigate to Businesses tab > Users
- Click on the checkbox to select the user(s) > click on 'Bulk Update x users'
3. Under the 'Tab Access' settings > click the drop-down arrow next to ''Inbox Messages'' and select ''Turn Off'', then Apply Changes to Users.