Once the upgrade is done, test the forum. Nothing elaborate. Just see if things are working. If you have a lot of hacks installed, then it might be a good idea to see whether they all work. This is a lot easier to do if you have more than one administrator on the site!

You can spend as much (or as little) time as you want doing this. As a bare minimum, check that you can still log in, post messages, and read messages. If you have a busy forum rest assured that the members will soon let you know if other things aren't working!