I'm having the same issue. I think it's because I have the default routes off, but I can't find any reference to what the route should be. If I put the default routes back in just to test, it works.
Wait, I just checked the source. You should see something like this:
<span class="in_place_editor_field" id="page_text_1_in_place_editor"></span><script type="text/javascript"> //<![CDATA[ new Ajax.InPlaceEditor('page_text_1_in_place_editor', '/groups/set_page_text/1', {rows:1}) //]]> </script>
So, in your case, just make sure that /users/set_user_first_name/ is a known route and you should be good to go.
Sean
