If your category or archive page, or any other page is set to show excerpts instead of the full content the add this button will not show.
This is bad enough, but rather than just not showing, what happens is the raw code that powers the button shows. So no graceful degradation here.
The simple fix for this hat can edit their theme files: switch the_excerpt for the_content.
It is really disappointing when a company like addthis releases something half finished like this. When a freelance developer makes a plugin in his spare time, fair enough bugs are to be expected, but when a company releases a plugin with major flaws, thats just laziness and poor attention to detail.
This is part of addthis.com's business strategy: it ought to be done right.