Order by ASC, DESC, Title, Date from URL on custom page template post

When you run an additional query, WordPress isn’t going to automagically parse get vars, that only works for the main query. You have to fetch and add them manually.

Also, don’t use query_posts, use WP_Query instead.

$args = array(
    'category_name' => $tutcatname,
    'posts_per_page' => 25,
);

if( isset( $_GET['order'] ) ){
    $args['order'] = $_GET['order'];
}

$query = new WP_Query( $args );