Your code isn't looking for any variable in the URL/query string - it's looking for data POST'ed by a form (hence the word "POST" in the $_POST variable).
Now, if you were to use [man]$_GET[/man] on the other hand...
You might want to look at a few pages:
The manual definition of what a [man]string[/man] is and why your $path variable wasn't being parsed.
The manual page for [man]variables.predefined[/man] that talks about the different superglobals ($GET, $POST, $_COOKIE, etc.).
You might want to secure your code so that only files you want to be included are allowed. One common way of doing this is to create an array of possible values for the user-supplied variable and use [man]in_array/man to match what they gave you against your array of valid choices; only if you get a match should you include() the file.
The other method of securing your include() calls is to place all of the .php files you want included in a certain folder "includes/" for example, and use [man]basename/man to verify that they didn't give a path outside of that directory (e.g. "../badfile.php").