<p>My AS3-heavy project tests fine in the IDE, but some parts don't work in the browser when published. Any ideas what could be causing this?</p>

It might help if you provide some more specific information on what's not working.  Can you post a link?  Furthermore, are any parts of your project referencing any remote objects?  If so, you might want to make sure your remote objects are located where your app expects them to be.  You may also need to add a crossdomain.xml to your server root if you're doing any cross site scripting.