Why do you think you can't do it? because Virgin are spoil sports or because the content providers won't allow it
A little of both to be honest. Content providers should not have any issue with Virgin engineers being able to transfer entire recorded content over to a new box when upgrading or replacing a failing unit (which was the subject of this thread) which would be a trivial task as long as the original HD is still mostly viable. Assuming that Virgin wanted to spend the resources doing so for their customers anyway.
I can see some content providers not wanting copying to PCs or streaming to other devices though (although Sky seem to manage it with Sky Go?) as many of these companies still try to live in the dark ages of national borders and release windows and want total control over media consumption. It does all seem to be a case of locking the stable door after the horse has bolted though, as who on earth would bother waiting for UK broadcasters to eventually catch up in order to pirate something from a Virgin TiVo?! Sadly it's a case of penalising legitimate use in order to supposedly prevent something that has actually already happened.
Anyway, this is probably a discussion for another place, as the answer to the OP's question is still going to be No.