We're off again, and I've been onto support both at our hosts and at Invision, who supply the forum software. Each time the server goes off, we get it back up and running again reasonably quickly (providing I'm not asleep or elsewhere), but we never quite get to the bottom of the cause. This is what I'm going to try & do this time around. Wish me luck!
Quick update 5:30pm Sat 25 Jan 2014: Support are going to get the actual box in the data centre on screen to see what info they can glean from the box before trying another reboot. We're working on it, so bear with us :)
Late Update: I've disabled some 3rd party hooks (think add-on apps), and the last one to be disabled brought the load right down. I'm leaving it at that for now, and have passed some information to Invision for them to study. Thanks for bearing with us ;)