My email notifications don't update very well. They generally show up, but never disappear until the app is opened, even if they've been moved or deleted on the server.
For instance, most Junk mail gets a notification. If I've got mail open on my computer it goest to the Junk Mail folder almost immediately. Unfortunately the iPad notification stays until I open the Mail app and it syncs with the server.
I've had apparently stuck notifications before. They turned out to be mail waiting in a second account's inbox. If you've got multiple accounts just tap the back button on the top left (landscape view) until you can see all the inboxes.
Now I just leave All Inboxes selected, and that solves that.