Get page id by template

When a page is created, the assigned template to that page is saved as custom post meta in the same way as custom fields. The meta_key is _wp_page_template and the meta_value will be the page template

You can simply make use of get_pages to retrieve all pages which have a meta_value of the specified template

$pages = get_pages(array(
    'meta_key' => '_wp_page_template',
    'meta_value' => 'page-special.php'
));
foreach($pages as $page){
    echo $page->ID.'<br />';
}

EDIT 23-07-2015

If one just needs the page ids, then you make use of get_posts and then just pass page as post_type and ‘idsasfields` value. This will ensure a much faster, much more optimized query as we will only return the post id column in the db and not all of them for the given pages

(Requires PHP 5.4+)

$args = [
    'post_type' => 'page',
    'fields' => 'ids',
    'nopaging' => true,
    'meta_key' => '_wp_page_template',
    'meta_value' => 'page-special.php'
];
$pages = get_posts( $args );
foreach ( $pages as $page ) 
    echo $page . '</br>';