functions.php

<?php
  include 'remove-comments.php'

remove-comments.php

<?php

// Removes from admin menu
add_action( 'admin_menu', 'my_remove_admin_menus' );
function my_remove_admin_menus() {
  remove_menu_page( 'edit-comments.php' );
}
// Removes from everything
add_action('init', 'remove_comment_support', 100);

function remove_comment_support() {
    $post_types = get_post_types();
    foreach ( $post_types as $post_type ) {
        remove_post_type_support( $post_type, 'comments' );
    }
}
// Removes from admin bar
function mytheme_admin_bar_render() {
  global $wp_admin_bar;
  $wp_admin_bar->remove_menu('comments');
}
add_action( 'wp_before_admin_bar_render', 'mytheme_admin_bar_render' );

http://www.reddit.com/r/Wordpress/comments/2ztv6f/how_to_completely_remove_comments_in_wordpress/cpmj24g

To make this more "complete" you could change the remove_comment_support() function to the below, so that the change is applied to all post types, not just posts and pages:

function remove_comment_support() {
    $post_types = get_post_types();
    foreach ( $post_types as $post_type ) {
        remove_post_type_support( $post_type, 'comments' );
    }
}