In my experience (nearly 15 years in enterprise development) this isn't the case at all. It used to be the case, especially with apps that used plugins or Java (both of which were common in the early 2000s), but for the most part these apps have gone now.
I think it is very, very unlikely that any non-client-Java webapp that runs in Firefox 3.x will break in any significant way in any future release.
edit: Given how long mainframes have been around, is it really that much of a stretch to say that these kind of legacy web apps could still be around for a few years yet, if they're really integrated deep into an organization's processes?
Yep. Hence my comment "I think it is very, very unlikely that any non-client-Java webapp that runs in Firefox 3.x will break in any significant way in any future release"
IE6-only (and IE-only) exist in the enterprise. That doesn't really impact on the Firefox upgrade strategy, since people don't use Firefox to access them.
(Also, MS is slowly dropping support for operating systems supporting IE6. Enterprises are very reluctant to use unsupported operating systems.)
It would be pretty ironic, but thanks to HTML probably not that hard, to be screen scraping an IE only web app to make it available to other web browsers.
That's what makes them so hard to port - they aren't really webapps at all - just (old style, non .NET) client-server VB programs that happen to be deployed as ActiveX controls in a web page.
Instead of mandating everyone run IE6 from now to eternity, IT will provide a Citrix link to launch the crusty web app inside the controlled Citrix environment, and allow users to use more modern browsers by default.
Or so we can hope.