In "The Robots of Death", "Grimwade's Syndrome", a fear of robots, was named after director (and, later, writer) Peter Grimwade, who hated to do robot stories. The same story had a character named Poul, presumably after Speculative Fiction writer Poul Anderson. Given the robot theme, Uvanov in the same story may have been a slightly subtler reference to Isaac Asimov, and Taren Capel to Karel Capek. In the novel Corpse Maker, a sequel to "Robots of Death" by the same writer, Poul is given the first name Andor.
"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.!
In the 1988 serial "Dragonfire", 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 "Human Nature", 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".
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 Hitchhikers Guide To The Galaxy.
A similar reference was made in "The Christmas Invasion": "Not bad for a man in his jim-jams. Very Arthur Dent. Now, there was a nice man."
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 arive 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."
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?
"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.
In "Planet of the Ood" where the saleswoman talks about the Ood's voice modifications one of them says "Do'h" in a way that sounds like Homer Simpson.
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".
Of course, the Cybermen and Borg have obvious similarities, but they made it even more obvious when a Cyberman said "YOU WILL BE ASSIMILATED!" in the episode "The Pandorica Opens."
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.
"The Empty Child" and "The Doctor Dances" seem to contain a Shout Out to The Importance of Being Earnest with Jack shagging Algy, since Jack and Algy (short for Algernon) are main characters in that play.
Word of God has confirmed that the scenes of Amy and Rory running through the TARDIS while it is possessed by House in "The Doctor's Wife" are a "very intentional" reference to Harlan Ellison's "I Have No Mouth And I Must Scream." 
"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 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.
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.
"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.
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.
In "The Day of the Doctor", as the Doctors enter the TARDIS to return to the present day from the 1500s, the 11th Doctor says 'Right then, Back to the Future'.