My developer took care of the problem. We had to adjust the code:
<?php $downloadlinks = get_post_meta($post->ID, 'downloadlinks', $single = true);
if ($downloadlinks==''&& is_user_logged_in()){
$downloadlinks = print '<ul><li>There are no download links currently available.</li></ul>';
} else {
if ( is_user_logged_in() ) { $downloadlinks = do_shortcode($downloadlinks); }
};
?>
Not sure if this helps. Hope it does.