(Assuming you are working with the default theme:)
Delete wp-comments-post.php and wp-trackback.php in your root directory, and then delete comments.php and comments-popup.php in your theme directory.
Then, in the index.php in your theme directory delete:
<?php comments_popup_link('No Comments »', '1 Comment »', '% Comments »'); ?>
In the single.php of your theme directory, delete:
<?php if (('open' == $post-> comment_status) && ('open' == $post->ping_status)) { // Both Comments and Pings are open ?>
You can <a href="#respond">leave a response</a>, or <a href="<?php trackback_url(display); ?>">trackback</a> from your own site.
<?php } elseif (!('open' == $post-> comment_status) && ('open' == $post->ping_status)) {
// Only Pings are Open ?>
Responses are currently closed, but you can <a href="<?php trackback_url(display); ?> ">trackback</a> from your own site.
<?php } elseif (('open' == $post-> comment_status) && !('open' == $post->ping_status)) {
// Comments are open, Pings are not ?>
You can skip to the end and leave a response. Pinging is currently not allowed.
<?php } elseif (!('open' == $post-> comment_status) && !('open' == $post->ping_status)) {
// Neither Comments, nor Pings are open ?>
Both comments and pings are currently closed.
Also delete:
<?php comments_template(); ?>
That should be it.