Thanks for the suggestions beau420.
There isn't actually any problems logging in. I can log in using the app as normal and just ignore the notification. Clicking on the notification just takes you to a screen which says 'You have already authorised Sony Ericcson Xperia. Press Okay to continue', and when I press OK, then it disappears, only to reappear a minute or so later.
When I go into accounts/sync, there are two different facebook icons, which differ slightly. If you click on the slightly bigger icon (the regular facebook icon) it shows my email address at the top with the word 'Facebook' underneath, and a sync Contacts option. The other icon has 'Sony Ericcson Facebook Integration' under my email address with the options to sync contacts, calender and albums in the gallery.
If I 'remove account' on the regular icon, it does so and the other icon disappears too. But if I 'remove account' on the other icon, it just reappears again immediately. I don't know if this has anything to do with it.
I tried clearing the cache/data but that didn't work. I'm running 2.3 by the way, and never had a problem with it until today...