In general, I would say to install software on a test server first if making a mistake on the production server could clobber a production application. However, it sounds like you're installing this for the first time and that it doesn't affect anything already in production.
I don't know anything about installing Invision Power Board, but have you read through their installation guide? Perhaps that would clear up your questions.
Yes, you should always test any program! It does not mean there is anything wrong with the program per say, but the program may not work the same on one server running certain versions of MySQL, PHP! It is also a great time for you to learn some of the key features before putting the service on a public network!