It's one of the main things they dragged onto the show with the reboot, adding romantic elements to lots of the Doctor's relationships rather than them being merely platonic. It sometimes works but more often comes off as a bit creepy due to the enormous imbalance in the power dynamic due to the Doctor's age and experience. Arguably it works best between him and the Master since that is not an issue but guessing they might do it with Davros is hardly a leap.
The worst thing about the shipping in NuWho for me, is actually the pandering to the worst parts of the audience. I'm thinking in particular of Amy Pond, Rory and the Doctor 'triangle'. Early Amy was a fun character. And whilst the stories got a little more turgid over time she was rarely without fun moments. When we met her she was a pretty lively character, sexy and fun. And I remember very distinctly on Gallifrey Base (big Who fan forum) when the previews for S5 went out a lot of seething about the character - she had a really short skirt, for example. And when the first episode aired, a majority of that forum was having a meltdown about how much they hated her - focused most of all around this exchange:
"This is my friend Rory.
"Boyfriend"
"Kind of boyfriend".
To me, and in other circumstances most people, would see it as quite creepy that Rory corrected her and told her what the status of their relationship was when she evidently didn't feel the same way and so much of what we could see of them was that she'd never really fancied him and he'd sort of friend'd her into a relationship. All this came to a head with the end of an episode where Amy makes her move on the Doctor. Frankly, I found it funny. His reactions, her being the sort of girl willing to make the next move and how he'd never really had to deal with that before. And honestly in every single respect he WAS the better man than Rory - she was quite right in finding the Doctor more interesting and suitable.
Gallifrey Base was this tiny volcano of impotent rage at her.
And then the Doctor turned her down and set about setting her up - quite manipulatively - with Rory. If anybody recalls the "Amy's Choice" episode where there are two parallel realities and he seemingly knowingly sets up this situation where Rory "dies" in very gruesome fashion right in front of her only for her to wake up and find Rory there. Whom the Doctor nudges forward to go and make his move. Yes, because traumatic shock and grief are exactly the right time for her to make a rational choice about the relationship, Doctor. But now, Gallifrey Base is all about it. Because she's with Rory and he's got the girl and she's learned her lesson.
That's the worst thing about the shipping in Doctor Who. I mean, you'd think it would be "Rooossse" and all that maudlin stuff and gods did that get old. But on the other hand, I actually liked the chemistry between Clara and Eleven (at the end of one episode, he smacks her on the arse with a chamois leather and she looks like she quite enjoys it). And unlike Rose or Amy, Clara (original Clara) was much more intellectually on his level. In the 'Snowmen' episode, there are lots of little intellectual flirtations between them - like where he's testing her to see if she'll figure out why they've come to the roof, why he picked up the umbrella, etc. and his delight when she does. Or her "It's smaller on the outside". Of course they couldn't leave well enough alone and had to fuck that up with later shipping her with that Norsewoman. Probably the most chemistry I've seen with the Doctor since original Romana (scorching!).
So all that is true but for me it's still topped by the sheer pandering to angry beta types in the Fandom like the fuming "Amy is a slut/whore/wont fuck my stand-in" types on Gallifrey Base.
