(Update 2: Classic Doctor Who is renewed too. Not that there's much of it on there.)
(Update: Turns out that shortly after posting this, Netflix has apparently renewed 21 BBC shows, including New Doctor Who and Torchwood. Go figure. This leaves the majority of BBC shows,
A few weeks ago, Netflix US put up a date on all the BBC shows, including Doctor Who, Sherlock and Torchwood, that they will expire February 1st. This has, for obvious reasons, caused an uproar in the fandom, spouting petitions everywhere, for example, one currently has 35957 signatures, with many sites fear-mongering the fans with click-bait headlines. The fans are showing that they are truly desperate for their show to remain.
In fact, the fear-mongering and petitions could damage the chance of shows returning and staying, in part or in whole, at worst. At best, it's pointless.
This is because this same thing happens almost every year, and not just with BBC shows, to varying degrees of panic. Hey, here's a petition from 2012 with the exact same thing, listing another previous occasion. Here's a discussion on Reddit from the same year about it. Here's a list of shows to be removed from last year.
The license Netflix has with the BBC's commercial subsidiary, BBC Worldwide, is up for renewal once again. Until the deal is closed, they need to deal with the worst case scenario where it falls through, so they put an expiry date up (and temporarily take it down if the negotiations go beyond that expiry date). Usually, negotiations are finished quickly so most of the time, people don't notice.
But how can public petitions damage the chance? Well, Netflix may want to try to get every show on the planet but they only have a certain amount of money to spend. BBC Worldwide is out there to make money. What do you think they'll do when they realise that fans, and therefore Netflix, are desperate for the show to stay? They'll put up the price! Especially since the show's popularity is soaring. This only gives WW more leverage.
This means that something will have to give. It may not be Doctor Who. It may not even be a show, but rather the technical setup to keep the site running smoothly (like battling American ISPs). But something somewhere would have to give. Unless, of course, Netflix deems it too expensive and simply rejects it.
So while it's important for BBC Worldwide to see a level of interest, keep the desperation in private with Netflix. Put your money where your mouth is and get on the phone or live chat with their support team, explaining how you need the show and would stop paying otherwise.
This way, each individual complaint can be tied to the issue and to money they could lose. This works infinitely better than an easily faked petition from people who could be members or not, and is one of the main methods Netflix uses to determine what shows are sustainable to keep.
Remember, be polite while contacting Netflix. The support staff aren't responsible for any negotiations and Netflix don't remove them because they want to, but legally have to.
So relax people. There's a good chance they might disappear for a short period, but it won't be gone forever (unless in the rare chance that negotiations fall through and if they do, other services exist). Chances are, the deadline date will just disappear and things will just continue. It has been confirmed Netflix UK and Ireland are unaffected.