By default Thunderbird only checks for new messages in the Inbox. If you use server-side filters to automatically move certain messages into folders, Thunderbird may not indicate the new messages until you click on the folder.
To fix this, in Thunderbird, select Preferences > Advanced > General > Config Editor... then in the search box, enter: mail.server.default.check_all_folders_for_new and double-click the value to set it to 'true'.