You just use a normal if statement around the notice so that it only appears if the value of your option matches the current date.
function wpse_277689_admin_notice() {
if ( get_option( 'expiry_date' ) === date( 'd/m/Y') ) :
?>
<div class="notice notice-success is-dismissible">
<p><?php _e( 'Done!', 'sample-text-domain' ); ?></p>
</div>
<?php
endif;
}
add_action( 'admin_notices', 'wpse_277689_admin_notice' );