What does it take to get FaceBook to stop pushing notifications to my iPhone?
Solution 1:
If there is something FaceBook is good at, it's in bothering it's users :)
So make sure you've unchecked those items:
Make sure you've switched the Notification Center-slider to off and the Alert Style to none in such a way FaceBook is listed in the Not In Notification Center.
And if you go to Settings < FaceBook, make sure you're Push Notifications are all switched off.
Solution 2:
Settings -> Notifications
You have to choose none in the notification style menu, and disable all buttons.