I've no idea. I cannot duplicate the issue on this end (and no, you do not have to be logged in for the function to work).
Using Windows XP SP3 with Google Chrome 3.0.195.33, Opera 10.10, Safari 4.0.2, IE8 and Firefox 3.5.5, everything works fine.
Are you sure the page has completed loading into your machine? If your machine is loading pages slowly, it might seem as if it is not hitting the last post. If your seeing the Done message in the lower-left corner of Firefox, then that wouldn't be it.
I dunno. Two ISPs, two operating systems, a handful of browsers in each and everything is working for me.
When you hover over that linked image, the URL should end with - /page_view_getlastpost
Is that what you're seeing?
Clicking on that link will resolve to a URL ending in - /10316-go-to-last-post-doesnt/page__pid__108913__st__0&#entry108913
Do you see that?