I have a site that, once an Admin is logged in, clicking the “Visit Site” menu item takes me to the home page of the site and makes it look like I’m not logged in. I assume this is because W3TC is serving the old, un-logged in page.
So “yes” to your question, in my experience.
This isn’t the desired behavior, of course. No idea how to work around this, so I disabled W3 Cache for now 🙁