How to remove some item from WordPress Dashboard for user Author

Two edits: checking for a role is not recommended, you want to check for a capability instead; and the php file you want to pass is edit-comments.php.

<?php
add_action( 'admin_menu', 'my_menu_links_removing', 999 ); 
function my_menu_links_removing() {
    if ( ! current_user_can('activate_plugins') ) {
        remove_menu_page( 'edit-comments.php' );                
    }
}
?>

Administrators are the only default role (besides Super Admin if you’re on MultiSite) that can activate plugins, so this still checks to see whether the user is an admin, and if not, the comments menu should disappear.

See also: https://codex.wordpress.org/Function_Reference/remove_menu_page#Examples for a list of other menu pages that can be removed and what their php filenames are.