Add Taxonomy Values Within a Custom Post Type RSS Feed

get_post_meta is only for custom fields. If listing_category and listing_type are in fact taxonomies, you need to use get_the_terms instead. The resulting code would be something like this:

add_action('rss2_item', 'yoursite_rss2_item');
function yoursite_rss2_item() {
    if (get_post_type()=='listings') {

        $fields = array( 
            'listing_bedrooms',
            'listing_city',
        );

        $post_id = get_the_ID();
        foreach($fields as $field) {
            if ($value = get_post_meta($post_id,$field,true)) {
                echo "<{$field}>{$value}</{$field}>\n";
            }
        }

        $taxonomies = array(
           'listing_category',
           'listing_type'
        );

        // Loop through taxonomies
        foreach($taxonomies as $taxonomy) {
            $terms = get_the_terms($post_id,$taxonomy);
            if (is_array($terms)) {
                // Loop through terms
                foreach ($terms as $term) {
                    echo "<{$taxonomy}>{$term->name}</{$taxonomy}>\n";
                }
            }
        }
    }
}

See the documentation for get_the_terms here: http://codex.wordpress.org/Function_Reference/get_the_terms