This is a new issue I think. Using the new dialogue box, if you open then decide you don't want to put anything in so alt+f4 to exit it, I think that's closing zotero completely within word such that clicking any of the add-on buttons does nothing. I've replicated a couple of times (accidentally). The dialogue box either needs a cross on it, or/and for the behaviour to change (if I'm right about what's happening - I'm open to alternative interpretations).

...I also can't see how to 'restart' it without restarting firefox (just doing word doesn't seem to do it).