It looks like you have an issue with your .htaccess file or your mod_rewrite configuration on your server.
If you try to access the WordPress ajax handler on your server, it should print a blank white page with a 0 or -1 printed to it. However note, that it can not be reached. This file should be publicly accessible, as WordPress does use it for background processes, as well as plugins for ajax requests.
Usually it is due to a permalink misconfiguration, or a cache plugin related misconfiguration in the the server .htaccess file, or something with a firewall.