By default, you only get notified when you have new mails arrived in your inbox. If you have a rule set up in your mailbox that skips certain emails to one of your sub-folders, you won’t get notified on your iPhone or iPad when these emails arrived.
Here is how you can change it to have these sub-folders also enabled for notification.
Go to Settings → Mails, Contacts, Calendars → Fetch New Data
Tap the mailbox you want to have sub-folders set for notification, and select them.
Go back and it’s all set. Next time when an email arrives into one of these sub-folders you will get notified right away.