This appears as a fieldset I think - on a couple of files at least.
Start with:-
includes/templates/template_default/templates/tpl_checkout_shipping_default.php
Make a COPY of this on your hard drive.
Find (probably close to the end of the file) the section:-
HTML Code:
<fieldset class="shipping" id="comments">
<legend><?php echo TABLE_HEADING_COMMENTS; ?></legend>
<?php echo zen_draw_textarea_field('comments', '45', '3'); ?>
</fieldset>
and COMMENT it out (ie: use <!-- and --> tags):-
HTML Code:
<!--<fieldset>
<legend><?php echo TABLE_HEADING_COMMENTS; ?></legend>
<?php echo zen_draw_textarea_field('comments', '45', '3'); ?>
</fieldset>-->
SAVE the file.
LOAD the edited file to your OVER-RIDES directory:-
includes/templates/YOUR_TEMPLATE/templates
You will need to find similar references on other tpl_xxx.php
Look for the ones with tpl_checkout_xxxxxx.php
Again, work on a COPY of the file. Then FTP the EDITED file to your over-rides directories.
TEST THE WHOLE CHECKOUT PROCEDURE! INCLUDING THE E-MAIL CONFORMATION! If my suggestions mess things up, REVERT to the original files (which should be safely untouched in their original directories.