Go into settings and notifications you might find an on off switch.
Gina7239 provided a great first step. When you open the Facebook application press Menu>Settings>Check Notifications (Active) and all the notifications you want to receive such as Messages, Request and Invites>Check the ringtone you desire and or LED option. Are the notifications working for a while and suddenly stopping? If so, please navigate to the phone Settings>Applications>Manage Applications>All>Facebook and clear data and cache. Power the phone off and on and tap the Facebook application and sign in. Please observe the notifications and advise if this was successful for you. Thank you.
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