Facebook's like buttons use metadata that are embedded in the HTML markup of a single page. It would work in theory if your URL's produce different metadata when accessed by curl (i.e. without JavaScript, which I believe is not possible since the hash portion is not seen server-side, so there is no way to act on hash string values server side).
So having said that, I would assume that Facebook Like buttons don't behave differently for different hash strings. A look at the facebook documentation (which mostly sucks) doesnt mention this, but the facebook developer forums seem to confirm this: http://forum.developers.facebook.com/viewtopic.php?pid=240151
However, all is not lost. You can specify the URL of a like button, so just set the URL to be the SEO-friendly URL of the same page: http://developers.facebook.com/docs/reference/plugins/like
UPDATE - Clarifying from the comments
So, now that we know you have static versions of the hash string URLS. Before, you were most likely placing like buttons on the page with this code:
XFBML:
<fb:like show_faces="false" width="450"></fb:like>
Instead, you should be specifying the SEO version of the URL. For example, when you're on mysite.com/articles#story4, the code should look like:
XFBML:
<fb:like href="mysite.com/articles/story/4" show_faces="false" width="450"></fb:like>