As someone who has spent many weeks with a 3kb/s download limit, and has a very limited knowledge of coding, I can tell you that what you're suggesting won't work.

Adding a box doesn't decrease loading times, it INCREASES them. Because you might not see what's in the box, but it is being loaded. So you're actually loading the entire page + something extra.

So with 3kb/s, you'd be lucky to see the banner before your browser gives up on loading.

This idea would be a bit difficult to implement so I'm gunna say no, it would require the site to record what sections have been minimized and I don't know enough about phpBB3 to create this. You can however view an individual category by clicking the title, so for example clicking "Multimedia" on the index page will hide all the other sections and show only Multimedia