Okie dokie folks, and please let me know if this works for you...
To clear the Facebook API Error...
1) Open Facebook on a COMPUTER - you can try it on IE Mobile in Desktop Mode, but clicking the remove App link is almost impossible...
2) Go to the App Center section in settings (in the beta release with the gray bar on the left, this is found in the top section) and select MY APPS from the list on the left.
3) Find the Windows Phone App and remove it (press the little X in the corner to delete the App).
4) Go it to settings on the phone - select accounts - tap Facebook and re-enter your password - DO NOT DELETE YOUR ACCOUNT!
5) Change your profile photo using the Me Tile as usual.
This looks like a slight API change in Facebook that the App needs 'permissions' to change again. Deleting the App on the computer will NOT remove it from your phone, but you will need to go in and re-enter your password before it syncs again.
Again, would love to know if this works for everyone!
It looks like this clears the error... Allows the image to change, but then it reverts back to the original issue after it syncs with Facebook... Super strange!