The resource limits are all about CPU usage; the information you need is on the Wiki
High Resource Usage page.
That said, phpBB is not a notable CPU hog out-of-the-box. Like any database-backed dynamic site, it does do a bit of work whenever it has to render a page or add user content, but as such things go it's not a particularly bad actor. (There are more efficient forums than phpBB, certainly, but a vanilla install of phpBB is at least runnable in the Free Hosting environment.) There are plugins and themes, though (notably a popular CAPTCHA plugin and most chats) that
are CPU hogs that will use 100% every time they're invoked, and run afoul of limits in almost all shared hosting plans.
I'd suggest searching for "phpBB CPU usage" without the quotes and seeing what you can do with your installation before upgrading your service. If you simply upgrade and are using any components that will use up all of the resources in your new environment, you will have spent money for nothing. If you can get your resource usage below the limits without upgrading, you'll have done yourself a favour, and if you find that you are only occasionally tripping the limits when you have a lot of users aboard, you can make a more informed decision as to your actual server requirements.
Illuminated will give you a bit more headroom, but if you're running badly-behaved plugins that you can't do without, or if your forum is very heavily used, you may still run into problems. If that's the case, then there's really no alternative to putting your site into a hard sandbox—a virtual or dedicated server where it's impossible to go beyond your share of the resources.