In the plugin, qa-share-layer.php at line 76 you have to change the code from:
'facebook'=> '<iframe src="//www.facebook.com/plugins/like.php?href='.qa_path_html(qa_q_request($this->content['q_view']['raw']['postid'], $this->content['q_view']['raw']['title']), null, qa_opt('site_url')).'&send=false&layout=standard&width=53&layout=button_count&show_faces=false&action=like&colorscheme=light&font&height=20&appId=170382616390886" scrolling="no" frameborder="0" style="border:none; overflow:hidden; width:53px; height:20px;" allowTransparency="true"></iframe>',
to:
'facebook'=> '<iframe src="//www.facebook.com/plugins/like.php?href='.qa_path_html(qa_q_request($this->content['q_view']['raw']['postid'], $this->content['q_view']['raw']['title']), null, qa_opt('site_url')).'&send=false&width=53&layout=button_count&show_faces=false&action=like&colorscheme=light&font&height=20" scrolling="no" frameborder="0" style="border:none; overflow:hidden; width:53px; height:20px;" allowTransparency="true"></iframe>',
The layout parameter was given twice and possibly causing the problem. So remove "&layout=standard". (And appid in the end, is no valid parameter as far as I could find out).
@Noah: Maybe you can update this at github?
Edit: D+mn, again not working. I think we have to wait for facebook as mentioned in the stackoverflow link...