bbPress.org

not quite sure what you mean by ‘archived’ content

bbpress has statuses of

public,
private
closed
spam
trash

spam and trash depend on the user capability – so keymasters and moderators see these, but not participant level.

‘closed’ in bbpress just means that the area cannot have new things added.

If you want to have an old forum that is not seen by users, then make it hidden.

untested, but this should do it

add_filter ('bbp_after_has_search_results_parse_args', 'my_function')

my_function ($args) {
	
		// bbp_get_closed_status_id() REMOVED,
		$post_statuses = array(
			bbp_get_public_status_id(),
			bbp_get_spam_status_id(),
			bbp_get_trash_status_id()
		);

		// Add support for private status
		if ( current_user_can( 'read_private_topics' ) ) {
			$post_statuses[] = bbp_get_private_status_id();
		}

		// Join post statuses together
		$args['post_status'] = implode( ',', $post_statuses );
		
return $args ;

}