There's a bunch of individual bugs about this, which I'll try to find and link to this one but:
In Moodle Themes you have layout files that control the outer HTML of your site. If you change certain classnames then a variety of AJAX interactions throughout the site will silently fail.
Some possible solutions to this issues:
1. prominantly tell people that they aren't supposed to make any change the layout files in their theme.
3. fully document all the dependencies so that themers can avoid accidentally breaking things.
You need #page-header and #page-footer for PDF (etc.) embedding to work as intended.
Hiding and showing activities via the eye icon on the course page can be broken too.
The Activity chooser window that replaces the two drop downs will not become an active link unless the correct CSS is found.
I personally would prefer some combination of the last two. I think most of the dependancies are attempts at optimisation that don't provide enough benefit to counterbalance the unexpected breakages as things change.
I'll try to find the other bugs that document these in detail, but for an overview: