I understand how to use /etc/drush/drush.ini; however, I don't know what exactly to put in there in order for Drush to use a different temp directory when creating an archive-dump.
You can change drush temp directory by seting the TEMP environment variable.
will instruct drush to store your temporary dump to /mnt/tmp.
Depends on operating system, it can be set with different variable. For more details, the code is here: drush/includes/filesystem.inc: function drush_find_tmp()
For Drush 7 sql-sync you can specify --target-dump
That specifies the actual file Drush will write to on the target.
$ drush sql-sync @site.prod @site.self --target-dump=/tmp/self/prod-db.sql.gz
NB This should answer " Perhaps I should clarify a bit. I need to be able to have a drush archive-dump use a different directory than /tmp (which I think is set via an environment variable.) My /tmp mount point is too small to handle a drush archive-dump; so the dump fails. – Mike Broyles Mar 13 '13 at 0:34", but I don't have reputation here so therefor a new answer.
For Drush 6, see https://drupal.org/node/1806416. That patch may or may not get backported to Drush5. See that link for current status.