// add filters for specific tags to the widget and shortcode
add_filter( 'related_posts_by_taxonomy_widget_args', 'related_posts_filters' );
add_filter( 'related_posts_by_taxonomy_shortcode_atts', 'related_posts_filters' );

function related_posts_filters( $instance ) {

	// add filter to wp_get_object_terms
	add_filter( 'wp_get_object_terms', 'specific_related_tags', 10, 4 );

	// remove filter after the tags are filtered
	add_filter( 'related_posts_by_taxonomy', 'remove_related_posts_filters', 10, 4 );

	return $instance;
}

function specific_related_tags( $terms, $object_ids, $taxonomies, $args ) {

	// term ids you want to include
	$include_terms = array( 55, 34, 40 );

	if ( !empty( $terms ) ) {
		foreach ( $terms as $key => $term ) {
			if ( !in_array( $term, $include_terms ) )
				unset( $terms[ $key ] );
		}
	}

	return array_values( $terms );
}

function remove_related_posts_filters( $results, $post_id, $taxonomies, $args ) {

	// remove filters after the related posts are retrieved from the database
	remove_filter( 'related_posts_by_taxonomy', 'remove_related_posts_filters', 10, 4 );
	remove_filter( 'wp_get_object_terms', 'specific_related_tags', 10, 4 );

	return $results;
}