The Doctor's babbled dialogue upon regaining consciousness after regenerating in "Robot" is composed of slightly mangled lines from "The Time Warrior" and "Invasion of the Dinosaurs".
In "Pyramids of Mars", the Fourth Doctor accidentally calls Sarah "Vicki", referencing one of his first incarnation's companions.
In "Earthshock", the Cyber Leader plays clips of the Doctor's previous encounters with them.
The High Council of Time Lords makes reference in "Arc of Infinity" to the Doctor leaving Romana behind in E Space.
In "Terminus", Turlough inherits Adric's old room, last seen in Part One of "Earthshock", the story in which Adric was killed off.
According to Peri in "Attack of the Cybermen", the Doctor has recently referred to her as Tegan, Zoe, Susan, Jamie, and Zodin. The Doctor remarks that he hasn't thought about Zodin in years.
Seven gives Ace an ID card. Ace: "Who's Elizabeth Shaw?"
In "World War Three", when Harriet Jones asks the Doctor about his Northern accent, Rose answers (as the Doctor answered Rose when she first asked him about it), "Lots of planets have a north."
"Dalek": Several of the Doctor's enemies from the old show are on display in an underground museum.
"Bad Wolf": The Face of Boe is described as the oldest inhabitant of the Isop galaxy. The Isop galaxy was the setting of the William Hartnell story The Web Planet.
In the same episode, the security guards on the Game Station threaten to send the Doctor to the Lunar Penal Colony, a direct reference to the Third Doctor story Frontier in Space, in which the Doctor really does get trapped on the Colony.
And both "Bad Wolf" and "Parting of the Ways" had a transporter that left behind what appeared to be ash, echoing "The Twin Dilemma".
In "The Satan Pit", the Doctor mentions the inhabitants of Daemos, attributed in "The Daemons" as the source of demonic imagery in human mythology, among the races who share similar Satan-like characters in their mythologies.
In "Tooth and Claw", the Tenth Doctor arrives in 19th-century Scotland and uses "Dr. James McCrimmon" as his alias. Jamie McCrimmon was one of the Second Doctor's companions, hailing from 18th-century Scotland.
From the same episodes in Series 1, Jack Harkness mentions Pompeii and uses the phrase "volcano day." The Doctor uses the phrase again in "The Fires of Pompeii" when he and Donna figure out where they've landed.
At the end of "Turn Left", Rose's appearance and the upcoming danger is heralded by the words "Bad Wolf" appearing everywhere the Doctor looks. This is a shout out to the ongoing mystery of the first series, the repeated words Bad Wolf appearing everywhere they went, but not with nearly as much frequency.
Obscure one: In "Destiny of the Daleks" (Tom Baker era, late 70s), quoth Davros: "Doctor, do you believe your puny efforts can change the course of destiny? ... Destiny, Doctor.... Invincible necessity.... Power. My power. My invincibility. My supreme plan to control the universe... Errors of the past will be rectified. I will add new design elements to the Daleks' circuitry. They will be armed with new weaponry. Weaponry so devastating that all matter will succumb to its power! I will equip them with all the knowledge of the universe! ... The Daleks NEED ME!" — In "The Stolen Earth"/"Journey's End" at the end of season 30 (aka Series 4 in the new ordering), Davros does, indeed, equip them with a universal-knowledge level weapon capable of destroying, quite literally, all matter in the universe.
Probably the longest delay before a continuity nod, in "The Stolen Earth" the Doctor mentions, "Someone tried to move the Earth before, but that was a long time ago...", calling back to the Hartnell era in which the Daleks took over Earth and began drilling to its center to install a massive engine and move it to parts unknown (If you wondered, 45 years. 45 years!)
From the immediately following episode "Journey's End":
Davros: Impossible. That face...after all these years. Sarah Jane: Davros. It's been quite awhile. Sarah Jane Smith. Remember? Davros: Oh, this was meant to be. You were there on Skaro, at the very beginning of my creations!
Not as far reaching as most of the above, but in the finale of season 4, the Doctor and Rose ask if Gwen from the Torchwood team has a family history in Cardiff. She says that she does, going "back to the 1800s," indicating that the character Gwynneth from "The Unquiet Dead", played by the same actress, shared "spacial genetic multiplicity" with Gwen's family.
"Voyage of the Damned" nods to the previous two Christmas Episodes by having everyone realize that London isn't a good place to be on Christmas unless you like being at ground zero of an alien invasion.
So at the conclusion of The End of Time, the Tenth Doctor is about to regenerate into Matt Smith and say his last goodbyes to his companions over the years. And then there's Alonso from "Voyage of the Damned". And THEN there's the great-granddaughter of Nurse Redfern "Human Nature"/"The Family of Blood".
His good bye to Alonso is both nice and a bit thoughtless. He hooks Captain Jack Harkness up with Alonso. However, the scene becomes a lot darker if you know why Jack is there and not with Torchwood at the time.
In "School Reunion", Rose argues at companion from the 1970s (or the 1980s) Sarah Jane over who faced the toughest aliens and monsters, referencing actual episodes of the classic and new series, including the previous week's.
During The Sarah Jane Adventures, when she first encounters the Slitheen, Sarah Jane remembers that argument, saying aloud, "Slitheen in Downing Street". The Slitheen also mention their routine job where the family members never came back.
At the beginning of "Utopia," there's a mention of the events of "Boom Town" (the last time the TARDIS had to refuel in Cardiff) and the Ninth Doctor:
Martha: Wait a minute—there was an earthquake in Cardiff a couple of years ago. Was that you? Tenth Doctor: Oh, that was a long time ago. I was a completely different man back then.
In the next episodes continuing the story of "Utopia," the Master offers his wife a jelly baby. A bit of an Ironic Echo as well. the Master also mentions the Doctor fighting off Sea Devils and Axos, both from Jon Pertwee's tenure in The Sea Devils and The Claws of Axos.
The 11th Doctor's fez-and-mop combination mimics Seven in Silver Nemesis.
Quite possibly the biggest continuity nods are to the other doctors. The biggest one ever was when The Eleventh Doctor is facing the Atraxi in the Series 5 premier and asks them if the Earth is protected. The Atraxi looks at all the various monsters that show up in the series (including those that don't attack earth), then the Doctor asks "What happened to them?" Cue a twenty second montage of every single Doctor from William Hartnell to David Tennant with Matt Smith walking through the image of Tennant to say "Hello. I'm the Doctor."
In a DVD/Blu-Ray exclusive scene, the TARDIS gives Amy a sped-up slideshow of all previous companions. Upon seeing that almost all of them are attractive young women, Amy echoes five decades of out-of-universe potshots at the show by jokingly insinuating the Doctor's a Dirty Old Man. The Doctor, meanwhile, is irritated at the TARDIS for not showing Amy more wholesome examples like "the metal dog" (K9).
In "The Vampires of Venice", Eleven attempts to flash his psychic paper, but accidentally flashes his library card instead. His expired library card. Which expired so long ago, the photo ID on it is of the First Doctor, William Hartnell.
In "Let's Kill Hitler," when the Doctor activates the voice interface, the hologram shows an image of himself. He tells it to give it someone he likes, and it change to Rose Tyler, but that makes the Doctor guilty. It changes to Martha Jones- more guilt. It changes to Donna Noble - more guilt.
"The Day of the Doctor" has one that rivals the one in "The Eleventh Hour" not only do they show all the Doctors' but they're not in flashbacks, they're all there in person (Yet still stock footage) helping the current Doctors
In "The Time of the Doctor" The Eleventh Doctor uses "The Seal of The High council" The same one he stole from The Master in his Third incarnation. He even outright mentions that moment "longest plot device setup ever"
Also from "Deep Breath", the Doctor still isn't a hugger.
When the Teller is going through the Doctor's mind, he tells him to dig deep and see everything, mentioning scarf, bowtie and right now, which he claims was meant to be minimalist but turned out magician. (Much like how the florid Third Doctor was once mistaken for a carnie.)
The Spectre of Lanyon Moor features a character called Professor Morgan who was first mentioned in passing by Professor Amelia Rumford in the TV story The Stones of Blood as an archaeological acquaintance of hers. She particularly mentioned him with regard to Cornish fougos, a major plot point in this story. Both Professor Rumford in the earlier story, and Professor Morgan in this one speak of attending a Megalithic Symposium, and refer to a colleague called Leamington-Smith, who is apparently an "idiot".
In "The Kingmaker" features a scene in which a stranded Doctor leaves a note for his future self to find. It is "later" mentioned to have been delivered by "some bloke with big ears and a Northern accent", an apparent nod to Christopher Eccleston. In that same story, Tom Baker provides a cameo in the form of a clip from a history documentary, presented as the Doctor's taped notes (not actually Tom Baker, but a very convincing impression by Jon Culshaw).
Another character with an accent similar to Eccleston's also gets in a "Fantastic". Big Finish: tweaking the limits of BBC licensing restrictions for over 10 years.
And the fifth Doctor audio "The Whispering Forest" features the Green Crescent medical symbol from "New Earth". At the end, the Mara sarcastically asks if the Doctor is leaving Purity because it's "too green for you", referencing the final line of "Kinda".
All sorts of nods to past continuity (notably the Third Doctor era) were referenced in Who Killed Kennedy, where the plotline worked in many of the Doctor's adventures, from a ground level point of view. In his authors note, David Bishop acknowledged the Kurt Busiek/Alex Ross graphic novel "Marvels", which gave a ground level view of the Marvel Universe.
In The Clockwise Man, when Repple says his appearance was changed and he was exiled to Earth the Doctor says he knows the feeling.
Beautiful Chaos is a sequel to the TV story Masque of Mandragora, spinning from the Doctor's claim that the Helix would return after 500 years. It takes a moment to refer to the three other Expanded Universe media which took up this plot thread (a Past Doctor Adventures novel, a Sarah Jane audio drama, and a Doctor Who Magazine comic strip), explaining that they all involved "fragments" of the Helix, but this was the real thing.
In The Last Dodo, the Tenth Doctor names the titular bird "Dorothea", but never explains why. Dorothea "Dodo" Chaplet was a companion of the First Doctor.
In Millennial Shock by Justin Richards, the Doctor offers someone a jelly baby, and the person looks at the sweet in bewilderment. "It was a dolly mixture. He knew it was a dolly mixture." This is a reference to a scene in "Image of the Fendahl", where the Doctor offers a skull a jelly baby, while actually holding a liquorice allsort. (When a viewer pointed this out, Word of God was that the Doctor does this to confuse people.)
Subtle one: in the Eighth Doctor Adventures novel Seeing I by Jonathan Blum and Kate Orman (set on an Earth colony in the 23rd century) the Doctor is in a prison/mental institution called the Oliver Bainbridge Functional Stabilisation Centre, where people who Know Too Much are held. At the end of Blue Box by Orman (set in 1980s America), we're told that the villain ended up in the Bainbridge Hospital, which appears to be the same thing, only run by the US Government instead of a megacorp.