Is it possible to reply to Facebook event invites without having to log into Facebook? Every time I log into Facebook, there is a chance that I will get distracted and waste time.
I have been looking at this for a few days and the nearest thing I could find to what you want is an iPhone App called EventsBook
It can sync Events and Birthdays with Facebook although it is not clear whether you can accept or decline event invitations.
Short of writing your own Application it doesn't look like their is anything out there at the moment. ( I may write one myself - seems like a good idea! :)
It seems there is a service that provides what you want - it is called http://www.mixin.com/
It allows you to collate and manage events and calendar information from lots of various sources - including Facebook.
You can view a full list of features here - they look pretty impressive.
I have just registered and you can fully manage your Facebook events using the service.
Hope this helps!
You could create a Sikuli script that would periodically log in and answer invites. Sikuli is a computer-vision-based UI automation tool.
Check out the demos: http://sikuli.org/
EDIT: Something to look for -- they actually have some demos of folks using Sikuli to play Facebook games for them.