I noticed that although our Add To Any button was appearing, for some reason when hovering over the button the pop-up menu did not pop up.
I read number of forum posts describing a similar problem and while solutions such as clearing the cache or ensuring your template conforms to WordPress structure standards are completely valid, they didn't fix our issue.
Then I finally noticed that we had two instances of the Add To Any buttons on our page(s) - i.e. one above the post, and one below. When we removed one of the instances the menu began to work once more!
In conclusion, removing all but one instance on a single page was our solution, and may be yours too.
Hope this helps.