This is a very simple PHP issue, unrelated to WordPress. You need to check if the value is empty before echo
ing it. So, store them as variables first:
$twitter = get_the_author_meta('twitter', $author_id);
$facebook = get_the_author_meta('facebook', $author_id);
//etc
Then check them before displaying:
if(!empty($twitter)) {
echo '<a title="Follow me on Twitter" href="'.$twitter.'"><img src="twitter.png" alt="" /></a>';
}
if(!empty($facebook)) {
echo '<a title="Follow me on Facebook" href="'.$facebook.'"><img src="facebook.png" alt="" /></a>';
}
//etc