Harry's disguise in "Robot" is a Whole Costume Reference to Steed in The Avengers. Sarah Jane intentionally mixes up her pop culture when she sees him and compares him to... James Bond.
In The Robots of Death, "Grimwade's Syndrome", a fear of robots, was named after production assistant (later to be both director and writer) Peter Grimwade, who hated to do robot stories. The story also had characters named Ander Poulnote His first name is revealed in the sequel novel Corpse Marker, after Speculative Fiction writer Poul Anderson, Uvanov (Isaac Asimov), and Taren Capel (Karel Capek).
In this 1988 serial , the Doctor and a philosophically-minded guard debate the concept that "the semiotic thickness of a performed text varies according to the redundancy of its auxiliary performance codes." This is a direct quote from an academic media studies book entitled Doctor Who: The Unfolding Text, and roughly translates into laymen's terms as "the less relevant an in-joke is to the plot, the more cultural significance it has."
Almost every character in that story was named after someone famous to film history.
In the same episode, the Doctor entreats the Sycorax to leave the humans alone. He makes a heartwarming speech about how from the day they arrive on the planet, and blinking, step into the sun, there is more to be seen than can ever be seen... whereupon he remarks, "Wait, that's The Lion King."
Army of Ghosts sees the Tenth Doctor donning a "triangulation kit" consisting of a power-pack and three TARDIS roundels; the finished getup resembles the Ghostbusters' infamous Proton Pack. Cue a brief rendition of Ray Parker, Jr's classic theme tune.
The Shakespeare Code: New companion Martha remarks that the apparent witchcraft being done "all sounds a bit Harry Potter", The Doctor enjoins her to wait till she reads the 7th book (which hadn't come out when the episode aired). Later, they and Shakespeare have to, on the fly, say the right words to banish the Carrionites:
Shakespeare: Banished like a tinker's cuss, I say to thee... (looks to the Doctor) The Doctor: Uhh... (looks to Martha) Martha: Expelliarmus! The Doctor: Expelliarmus! Shakespeare: Expelliarmus! (it works) The Doctor: Good old J.K.!
The Doctor's human alter-ego gives his parents' names as Sydney and Verity. Sydney Newman was one of the original creators of the show, and Verity Lambert was its first producer... they could very well be considered the "mother and father" of Doctor Who! This is reprised in "The End of Time", the 2009 Christmas special, with a cameo appearance by one of Joan Redfern's descendants, who has written a book based on John Smith's journal under the name "Verity Newman".
To say nothing of the number of Agatha Christie titles dropped into the dialogue ("Well, that's put the Cat Among The Pigeons", "It's a trick, an illusion, They Do It with Mirrors", "The Moving Finger points to...", "Why Didn't They Ask - Heavens!")
Word of God is that the episode was inspired by the board game Clue ("Cluedo" in the UK).
Professor Peach: (in the library) I say! What are your doing with that lead piping?
In The Eleventh Hour, The Doctor's rejection of one food after another is reminiscent of the scene in Winnie the Pooh in which Tigger says "Tiggers like honey/yuck, Tiggers don't like honey." It even sounds similar: "Oh, I love apples" — *chomp* "Yech! I hate apples." It seems to fit the Doctor's personality as a bumbling, manic, impulsive genius.
During The Beast Below, Amy runs around in her nightie and a dressing gown the whole time, a bit like a female version of Arthur Dent. Or a reference to Wendy from Peter Pan; a girl who runs off with the mysterious boy to another land in her nightgown the night before she has to "grow up".
The sequence in The Time of Angels with the Weeping Angel slowly emerging from the television screen in is quite possibly a reference to The Ring.
Asylum of the Daleks has a possible reference to Coupling when Rory asks, "What colour?" because all the good questions were gone. Similarly, in "The Man with Two Legs", Patrick asks Jeff — who has fallen in love with a girl's leg — "Left or right?" When Steve stares at him, he protests "It's a leg! What else is there to ask?" The others look shocked at him when he queries about the colour. It's possibly a reference to the backlash against the brightly coloured Paradigm Daleks in Victory of the Daleks.
The Rings of Akhaten contains an array of new species. One of these seems to be a reference to CHIKARA, as not only are they called the Ultramanti (meaning the singular is Ultramantis), but there is a resemblence between them and one UltraMantis Black.
Deep Breath's restaurant scene is similar to the ones in both Sherlock Holmes films: In Victorian London a highly intelligent, gorgeously-costumed girl meets an eccentric genius in a restaurant who proceeds to insult her (they make up later). Then, it turns out all the patrons were in on the bad guy's plot; they get up simultaneously and leave the girl to her fate (though Clara fares better than Irene).
Psi seems like a fusion of several 90s cyberpunk tropes: His memory powers recall Johnny Mnemonic, and his stuttering is very reminiscent of Max Headroom.
In an episode that heavily features psychic abilities and the threat of fire, the main antagonist is named Delphox.
You've got an "Architect", someone who can change their face, a conversation about how impossible it is to not think about something, and the phrase "You'll be old and full of regret". Sound familiar?
Saibra is a mutant, whose powers allow her to assume another's personality and likeness via touch. However, this also dooms her to never being able to touch another living being as she can't shut the powers off. Tweak the powers a smidge, and give her a Skunk Stripe, and you have Rogue of the X-Men.
Clara's outfit. It's a simple black and white suit with a thin black tie. Being worn for a bank heist. Still haven't put it together? She was wearing it for her date with a man who could easily be called... Mr. Pink.
Although it isn't seen that way in the episode, the angle from which the Trafalgar Square lion's head is seen in the episode's main publicity shot (see above) makes it look a lot like the face of Swamp Thing. Which is appropriate given the resemblance between the fairy-dust entities and "the Green" in Swamp Thing.
The New Series Master feels like a Shout Out to Caligula in I, Claudius. He has a constant sound of drums in his head like Caligula has a sound of galloping in his head. Also the aged Doctor seems similar to Claudius who is mocked by Caligula. Suitable as in "Last of the Time Lords" the Master is The Caligula. If you watch the two characters you really feel the similarity between the John Simm Master and Caligula. John Simm had already played Caligula and claims he partially based his performance on this.
Although, ironically, the Master's previous incarnation was played by Claudius.
River's relationship with the Doctor, a woman who falls in love with a time traveller and meets him in the wrong order throughout her life is a pretty obvious shout out to The Time Traveler's Wife.
There's also a suspicion that Van Morrison is getting a namecheck here: reference his ballad Crazy Love (and what sort of love could be crazier?)
yeah I run to her, like a river song... she gives me love, love, love, love - crazy love!
The 2003 Doctor Who webcast "Shada", based on a script by Douglas Adams, included a Nutrimat drinks machine and a Ford Prefect car (neither appearing in the original script), both references to The Hitchhiker's Guide to the Galaxy. The 2012 novelisation contains several more references, such as a reiteration of the famous digital watches joke, Chronotis switching out the book for "an Earth classic about space and thumbing lifts..." and a scene where the Doctor attempts to communicate with a man in ragged clothes and long hair whose first word is, for no apparent reason, "It's..."
The title of the Doctor Who Expanded Universe novel The Taking of Chelsea 426 is a double reference: first (and most obviously) to the movie The Taking of Pelham 123, and second to the planet LV-426.