My WordPress installation isn’t updating the feed when a new post is published

Do you refer to feed settings in WordPress or W3 Total Cache?

Since I think this is indeed likely to be caching issue I’d check following:

  1. Check that feed is not getting stuck in browser cache (force reload, clear cache and reload).
  2. Verify that headers on feed make sense, http://redbot.org/ is good tool.
  3. Try to disable feed caching in Performance > Page Cache and see if it makes a difference.
  4. Enable debug mode for page cache in Performance > General Settings and look through information it gives on feeds.