There are import / export tools for a lot of the control panels if that's what you are using.
It's not that hard to move a forum from one server to another if you don't change the paths too much (though tbh this shouldn't affect you unless you've stuff hardcoded in, which is a bad idea in the first place)
Simply copy the files from server 1 > server 2
Take a dump of the database and import it into the new server.
You could probably do the entire thing in about 15 minutes (or less)
I've had to do this a few times over the last couple of years!