Retrieve a post’s featured image with PHP outside of WordPress

The most common way to achieve those features is to implement some sort of interface, usually in JSON or XML.

To do this, you either use a plugin (http://wordpress.org/plugins/json-rest-api/ <= Will probably part of core in WP 4.0) or do the AJAX function yourself.

An example would be:

add_action( 'wp_ajax_get_thumbnail', 'ajax_get_thumbnail' );
add_action( 'wp_ajax_nopriv_get_thumbnail', 'ajax_get_thumbnail' );
function ajax_get_thumbnail() {
    $post_id = $_GET['post_id'];
    $thumb = wp_get_attachment_image_src( get_post_thumbnail_id($post->ID), 'post-thumbnail' );

    echo json_encode($thumb);

    die(); // this is required to return a proper result
}

The call for that action would be: http://www.yourwebhost.com/wp-admin/admin-ajax.php?action=get_thumbnail&post_id=15

Please mind, that the 15 is just an example. You would get a json object in return, that contains witdh, height and the url to the post thumbnail.