It is not free for me in the UK to watch Top Gear or Dr Who, I am legally bound to pay a TV licence. I have no option but to pay this if I own a TV set, even if I don't watch the BBC and just watch DVDs I still have to pay the licence.
No you don't.
LinkNor do you need one to watch iPlayer programming after the show was originally broadcast (see link at the bottom of that page).
So if you wanted to watch Top Gear or Dr. Who in the same way that Tony does so (i.e. after it has been broadcast, even if only by an hour or two) you could do so legally for 'free' (i.e. without needing a TV license)