We have mailbox on exchange 2013 & using outlook 2013 at desktop & outlook/mail at iPhone & ipad.
In outlook (on PC or Mac) with an exchange server, you can set rules to place certain emails in certain folders. E.g if an email comes from Apple Support Communities, it goes into the Apple Folder within outlook. Now unlike Mail app, Outlook saves these rules to the server, so the email is NOT moved locally into the folder, it is done at the server level.
With mail on the iphone, any email that is in the Inbox will show an icon that states you have new email, however, if an email has been moved on the server to another folder (e.g Apple) then mail on the iPhone will NOT show a notification that this folder has new mail. In fact, you could have 100 new mails all in different folders and mail on iPhone will never indicate that you have new mail.
The only way to see if there is new mail, is to go into each folder manually, and then do a refresh on that folder in which case the mail gets picked up.
Many many corporations use Exchange server, and many of the employees have server side rules, i cannot for the life of me work out why mail on the iPhone is unable to identify when a folder has new mail.
The apple mail app & microsoft outlook app for iOS is unable to show unread emails at one location even if those mails are in the subfolders.
Please suggest some appropriate app or a way out by which I can accomplish the task.
Links to read: https://www.msoutlook.info/question/763
Other potential suitable apps for iOS: