Feeds module will not unpublish old feeds
I am looking to solve the same problem. So far the best I have found is a patch for the Feeds module: http://drupal.org/node/1240366

EDIT: That was a poor answer, because I did not test it. Also there is a long discussion in the "Sync or Cache mode" issue: http://drupal.org/node/661314 . Posts #114 and #119 have the two leading solutions (and #114 is more sophisticated).