There is no direct way to remove it from the shopping cart or billing screen. There are styles defined for the fields so you could probably change the css so the text doesn't display.
<td class="giftcert">Presented To:</td><td class="giftcert">jim</td></tr>
<tr><td class="giftcert">From:</td><td class="giftcert">jim</td></tr>
<tr><td class="giftcert" valign="top">Message:</td><td class="giftcert">jim</td></tr>
<tr><td class="giftcert" valign="top">E-mail:</td><td class="giftcert">
some-email@somedomain.com
I don't know css that well but you can probably define td for giftcert to be empty except on the screen where you want it to display. Maybe some css guru on the form could help with that.