Because /demos/hangman/index.php/game/guess/D is not a CGI appliction. POST requests can only be done to a CGI application.
/demos/hangman/index.php is a CGI application though. But for Hiawatha to search for a CGI application inside an URL, you have to switch on PathInfo. Use the following setting and try again.
VirtualHost {
...
EnablePathInfo = yes
}