Getting comment count per post not working

Try this:

<?php $popular = new WP_Query('orderby=comment_count&posts_per_page=5'); ?>
<?php while ($popular->have_posts()) : $popular->the_post(); ?>
    <li><a href="https://wordpress.stackexchange.com/questions/108038/<?php the_permalink(); ?>"><?php the_title(); ?></a> has <?php comments_number( 'no responses', 'one response', '% responses' ); ?>.</li>
<?php endwhile; ?>

From the codex.

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