From what I've experienced, fillins and asks pop up at open and save events,
and whatever passes for a "print event" in word, even if you've already
answered them & just want them to go away.

A couple of things come to mind, most of them involve macros or VBA
programming. You could, add a procedure to your print macro that delinks the
fillin fields prior to printing. This will remove the fillin field and leave
the text response as text.
Below are links that will take you to instructions on how to delink the
fillins in your footer, as well as what you would need to do to remove
fields in the body.

http://www.mvps.org/word/FAQs/MacrosVBA/UnlinkFldsFromHeaders.htm
http://www.mvps.org/word/FAQs/MacrosVBA/UnlinkFldsFromBody.htm

Have you put the \o switch on the ask field?
eg
{ASK  Name "Name"\o}

Hi Shirley,

If you create a template document with FILLIN or ASK
fields to prompt the user for info when creating a new
doc, then set the print options to Update Fields, this
causes the FILLIN and ASK fields to prompt the user to
answer the same questions again, or click cancel. To avoid
this use ASK fields configured as follows-

1. In the template, create ASK fields in the format:
{QUOTE{IF{BkMrk}= "" {ASK BkMrk "Question me if you will"
\d ""} ""}{BkMrk}}
where BkMrk is a bookmark named to match the particular
question.

2. Then, at the start of each field, create an empty
bookmark corresponding to whatever you call BkMrk for that
field. You can do this using a SET field (eg {SET
BkMrk ""} inserted before the "{QUOTE" or via
Insert|Bookmark (with nothing selected).

3. Save the template. If you created the bookmark via
Insert|Bookmark, don't answer the question before saving
the template, otherwise you'll have to delete the BkMrk
bookmark and re-create it.

Now, when the template is printed, users will only be
prompted to update any ASK fields they haven't answered.

Cheers
PS: Remove NO.SPAM from the above before replying.