Adding restrictions to open comments

All you need is to simply check if the entered name or email matched any admin accounts before the comment is saved.

function restrict_admin_names() {
    if ( !is_user_logged_in() ) {
        $name = $_POST['author']; // Get Submitted Name
        $email = $_POST['email']; // Get Submitted Email
        $admins = get_super_admins(); // Get an array of admin login
        $adminemail[] = ''; // Create an empty array of admin emails
        $admindisplayname[] = ''; // Create an empty array of admin display names

            // Create an array of admin emails and display names
            foreach ($admins as $admin) {
                $userdata = get_user_by( 'login', $admin );
                $adminemail[] = $userdata->user_email;
                $admindisplayname[] = $userdata->display_name;
            }


        if ( isset($name) && in_array( $name, $admins) && in_array( $name, $admindisplayname) ) {
            wp_die( __('Sorry, you can not use that name', 'mytextdomain' ) );
        }

        if ( isset($email) && in_array( $email, $adminemail) ) {
            wp_die( __('Sorry, you can not use that email', 'mytextdomain' ) );
        }
    }
}
add_action('pre_comment_on_post', 'restrict_admin_names');

Though it just checks for admins, but you can modify the code to check for any user roles or all the registered users.

Needless to say, this code goes into your theme’s functions.php file.