Comment_author_url doing nothing

Change your query from:

SELECT
    COUNT(comment_author_email) AS comments_count, comment_author_email, comment_author, comment_author_url
    FROM '.$wpdb->comments.'
    WHERE comment_author_email != "" AND comment_type = "" AND comment_approved = 1
    GROUP BY comment_author_email
    ORDER BY comments_count DESC, comment_author ASC
    LIMIT '.$amount

To:

SELECT *,
    COUNT(comment_author_email) AS comments_count, comment_author_email, comment_author, comment_author_url
    FROM '.$wpdb->comments.'
    WHERE comment_author_email != "" AND comment_type = "" AND comment_approved = 1
    GROUP BY comment_author_email
    ORDER BY comments_count DESC, comment_author ASC
    LIMIT '.$amount

And then this:

$output .= "<li><a href="https://wordpress.stackexchange.com/questions/108024/.$result->comment_author_url.">".$result->comment_author."</a> (".$result->comments_count.")</li>";

To:

$output .= "<li><a href="https://wordpress.stackexchange.com/questions/108024/.get_author_posts_url($result->user_id).">".$result->comment_author."</a> (".$result->comments_count.")</li>";

Modifying the query grants you to reach all fields, one of these ( user_id ), will be used to get the author’s page with get_author_posts_url();

Update

I suggest you to use double quote on SQL strings and single quote on HTML.
In your code you’re trying to concatenate $output with nothing.

So your queries might look like this:

"
    SELECT *,
        COUNT(comment_author_email) AS comments_count, comment_author_email, comment_author, comment_author_url
        FROM $wpdb->comments
        WHERE comment_author_email != '' AND comment_type="" AND comment_approved = 1
        GROUP BY comment_author_email
        ORDER BY comments_count DESC, comment_author ASC
        LIMIT $amount
"

Then for $output:

$output="<ul>";
foreach($results as $result) {
$output .= '<li><a href=""https://wordpress.stackexchange.com/questions/108024/.get_author_posts_url($result->user_id)."">'.$result->comment_author.'</a> ('.$result->comments_count.')</li>';
}
$output .= '</ul>';

Hope it helps!

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