Ashley is right, it will be customer/head.tpl
Edit this part:
{* <div class="line2"> *}
{* COMMENTING OUT CUSTOMER PHONES FOR FAST LANE CHECKOUT - NOT NECESSARY *}
{* {if $main ne "fast_lane_checkout"} *}
{* {include file="customer/search.tpl"} *}
{* {else}
{include file="customer/phones.tpl"}
{/if} *}
{* </div> *}
Put
{*
at the start, and
*}
at the end, of all the parts you want to comment out - they won't appear on your pages then.
In my case, I completely got rid of the .line2 div, as I don't need it at the moment.
I put a big capital letter comment in the code, so that I can see why I did it, if I come back to it in a few months' time! Otherwise you could be looking at something that is commented out, and not have a clue why you did it.