In my experience it is extremely important, especially for QDF terms (read up on “query deserves freshness” if you’re not familiar).
After all when we’re looking for information, we want it to be up to date — so all other things being equal, result that from last week is more appealing than a result from two years ago. Users want up-to-date information so Google tends rewards newer results. This has led to a culture of constant content updates across the industry.
In fact have also seen cases where a site ranks for a term with one page and simply updating a different page to have a newer datestamp than the ranking page results in that newer page taking the site’s spot for the term. So you’re not only competing with other sites from a freshness perspective, you’re also competing with yourself.
Good luck!