Is there anything us users can do to help? Does logging out (rather than staying logged in) help reduce the load on the servers? What about posting fewer twitter links and images etc?
Both of those things will help. We are discussing some stronger measures that will be hard for many users to deal with at first, but it will significantly improve the server load and allow us to get back to normal sooner.
1) Remove usernames and use a number system for each poster.
2) Prepare posting blocks - this means posters from say 1-300 can post during 12pm - 3pm, posters 301-600 from 3pm - 6pm etc.
3) Reduce post content size and introduce a character limit.
4) Encourage text speak as much as possible. For example "you stupid cnut" can be turned into "u dum cnut" which helps.
5) Stop responding to other peoples posts. If you're in an argument or debate with someone just post your message, say something like the above and leave it there.