A WebmasterWorld thread has Labnol asking, "Can rankings vanish because a site is not updated?" If you do not update a site's content, can it set off a red flag at Google or another search engine and have the site lose its current search rankings?
The consensus in the thread is that having stale content should not penalize the site. There are many sites that are rarely updated but continue to rank well for both competitive and long-tail keyword phrases.
It is always good to update and add fresh content to your site. Fresh content is useful for your end user, it encourages the spiders to visit you more frequently, which helps search engines pick up new pages quicker. It also gives you the opportunity to rank well for new keyword phrases, on those new pages you create. But if you decide your content is super and there is no need to update it because you rank well already, then not updating it should not impact your current rankings.
Everyone seems to be in agreement that there is something else at fault in this situation. Not updating content or having a stale site is not a reason for a downgrade in search rankings.
Forum discussion at WebmasterWorld.