Breaking is only part of the problem. The pages that rely on the referer header take it for granted and do not implement any meaningful error handling. They just die a horrible death, instead of responding with an error message stating that they need a referer.

One bad example is relying on the referer only for log-out, everything else works. That site also runs massive js on log-out, as if it really needs to rely on explicit log-out, and not just the user disappearing.