Nothing, and also everything
The definitive piece of writing on Doctor Who canon is this blog post by writer Paul Cornell. I'm essentially going to be summarizing his post here, much less eloquently, but one section I want to quote directly is this:
Nobody at the BBC has ever uttered a pronouncement about what is and isn't canonical. (As I'm sure they'd put it, being such enthusiasts for good grammar.) Because there was never a Who product that the BBC made that got a producer’s goat enough for that to happen. And because canonicity takes some explaining to anyone raised outside of fandom ('but… if it's got Doctor Who on the cover… how can it not be Doctor Who?') And because the continuity of Doctor Who was always so all over the place anyway that something in a new story not matching up with something from an earlier one was just the way things were, rather than an aberration that had to be corrected through canonical excommunication.
There is no absolute Doctor Who canon
The reason for this is largely historical; so let's take a brief divergence.
What exactly is "canon"?
The word "canon" comes to us from the church. Depending on precisely how you want to look at it, traditionally the word refers either to the body of laws of a religious authority, or to the set of approved religious texts. Precisely which is the true origin doesn't matter for our purposes, because they both have the same rule: only God (or an agent of God, like the Pope) can add to it. You need to have some weighty authority behind your words if you want to add to the Biblical canon.
Fast-forward to 1911 or so, when the word was applied to the works of Arthur Conan Doyle. The Sherlock Holmes stories were immensely popular in their day, and spawned a number of knock-offs. To distinguish between those and the Doyle originals, fans used the word "canon." Once again, you needed a lot of authority to add to the canon: only Doyle himself could do it, and once he died that was it.
Although we've expanded the definition in recent years to allow larger entities to issue pronouncements on canon (the Star Wars canon is now managed by an entire division of Disney, called the LucasFilm Story Group), those two basic principles have remained the same:
- The canon is the authoritative body on what "really happened" in a fictional universe
- You need some mighty powerful authority to add or subtract from the canon1
What does this mean for Doctor Who?
Authorial canon doesn't make a ton of sense for Doctor Who; who's the author? To put it another way, what makes Steven Moffat's word weightier than David Whitaker's?
The closest thing Doctor Who has to a consistent "author" is the BBC. They are the only group who can sensibly decide what is and isn't canon, and they don't want to. Without that pronouncement, nothing can be canon. And if nothing's canon, then everything is.
Having said that, fans generally agree that the TV show is more canon than the licensed products (the Big Finish audios and the novelizations). The main reason is because, for policy reasons, the BBC limits what the show can reference from non-televised material; in Cornell's words (from his blog post linked above):
BBC television dramas must be whole unto themselves, and must not require extra purchases that 'complete the story', as per the BBC charter2.
So there's an inherent imbalance between the canonical authority of the show over the licensed materials; Big Finish (and the rest) cannot add anything to the canon of the show, but the show can (and constantly does, because of course it does) add to the canon of the licensed works. The best we get is when the show adapts stories from the licensed materials, as with the two-parter "Human Nature"/"The Family of Blood" (based on Paul Cornell's 1995 Virgin New Adventures novel Human Nature).
This has come up in the fandom before; "Scream of the Shalka" was a BBC-licensed Doctor Who animated serial released online in 2003. Set after the 1996 movie, it starred Richard E. Grant as the Ninth Doctor. Obviously this was overwritten when the series returned to the air in 2005, and Grant's "Shalka Doctor" is no longer considered canonical by many fans (or, at best, he's an alternate canon - because Doctor Who can do that).
This is part of why the Paul McGann-featuring Night of the Doctor minisode was such a big deal to the fandom: it sort of skirted that rule (how, I'm not sure), and brought some of the Eighth Doctor's companions from Big Finish into the television canon.
The principal issue with not having a rigidly-controlled canon is that occasionally things contradict themselves. In some works that would be a problem, but Doctor Who's central conceit makes it a non-issue; as Stephen Moffat said at San Diego Comic-Con in 2008:
It is impossible for a show about a dimension-hopping time traveller to have a canon.
Ultimately, what's "canon" is what you decide is canon, in your own head, and nothing else. I have a different canon than you, and that's okay.
1 There are minor complications around things like fanon (with many definitions, but essentially non-canon elements that gain a certain degree of acceptance by the fanbase) and headcanon (personal additions or corrections to the canon), but I don't want to delve into those too deeply
2 I'm inclined to trust Cornell on this, because he's written for the BBC and I haven't; however, I have not been able to find a reference to this policy anywhere else; I'd appreciate a tip, if anyone knows where to find it.