I can't imagine trying to get everyone in a large corporation to use the right browser for the right thing. Most normal people barely understand that IE is not the internet, let alone understanding the difference between websites, web apps and internal and external servers.
In a very small company, maybe, but not in corpo-land.
There would certainly be a training issue, but I think people would get it if their system were configured so that each browser is only able to access its intended targets. Something as simple as giving the internal-only one a different, non-standard icon might help.
Using policies in IE6, it should be easy enough to disable access to internal sites. And in FF3.6 (for the internal apps) it's probably trickier (and I don't know how to deploy the policy), but one might point it to a non-existent proxy, and tell it not to use that proxy for internal sites.
In a very small company, maybe, but not in corpo-land.