Implementation of a URL router
Mainly I would like to know if I should be sanitizing any strings?


Assuming that the only unsafe input is in $_GET['url'],
you're safe: junk that doesn't match a defined route will be ignored.
The routes work effectively like a white list.


  How could I handle paths that don't match any defined routes?


Return a 404 error page (and correctly set status code in the header).



This seems quite redundant:


function __construct()
{
    if (isset($_GET['url']))
        $this-&gt;req_path = $_GET['url'];

    // Build the requested path.
    $this-&gt;req_path = is_null($this-&gt;req_path) ? "/" : "/" . $this-&gt;req_path;
}



This should be equivalent and shorter:

function __construct()
{
    // Build the requested path.
    $this-&gt;req_path = '/' . @$_GET['url'];
}




Instead of iterating over the list of routes and stop when you find a match,
why not use an associative array instead? Like this:

function route($path, $handler)
{
    $this-&gt;routes[$path] = $handler;
}

function run()
{
    if (is_null($this-&gt;routes))
        throw new RuntimeException("No routes have been defined");

    if (isset($this-&gt;routes[$this-&gt;req_path])) {
        $this-&gt;routes[$this-&gt;req_path]();
        return;
    }
}