Here's a few things I try when an app misbehaves and crashes.
1. Quit the app and restart it like this: Go to the Home screen and double click the Home button. That will reveal the row of recently used apps at the bottom of the screen. Tap and hold on the app in question until it wiggles and displays a minus sign. Tap the minus sign to actually quit the app. Then tap anywhere on the screen above that bottom row to return the screen to normal. Then restart the app and see if it work like normal.
2. If the first suggestion did not work try quitting the app again but instead of restarting it right away, reset your device like this: Press and hold the Home and Sleep buttons simultaneously ignoring the red slider should one appear until the Apple logo appears. Let go of the buttons and let the device restart. Ten restart the app and see if it works.
3. If neither of the previous suggestions worked I suggest deleting the app by tapping and holding on the app until it jiggles and displays a Rex X. Tap the X and confirm that you want to delete the app. Then reset your device. After the device has restarted go to the App Store and download Facebook again.
If all these fail you might just have to search for an alternative app until Facebook fixes the problem.