Look in Settings on your iPhone. Scroll down to Facebook, select it and then you will see Push Notifications. Select that and see if the settings are set to On.
Might be less the phone, more Facebook. My wife complains about it not working on her iPod touch a lot. Facebook in general is pretty quirky about working on it's own anyway.
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i think it works too well on my iPhone. way too much gets pushed to me. that being said, it was cool to see my whole address book get updated with facebook friend info and photos.