tl;dr: Yes, lots of them.
Writing “this was never acknowledged in the show” got quite dull, so I’ll say upfront that most of these repeated roles were never acknowledged or explained in the show. The exceptions are noted in the list below.
You can also branch off into the Big Finish adventures, other voice work, and so on. That makes the list even longer, and so I’ve not included characters like this unless they’re particularly significant in other ways (e.g. Carole Ann Ford).
Colin Baker, the Sixth Doctor, played a grumpy Time Lord called Commander Maxil in The Arc of Infinity, a story under the Fifth Doctor. His jovial nature on set was what marked him for the role, and the producer (John Nathan Turner) came back to him later when casting the Sixth Doctor.
Carole Ann Ford (who appeared in this evening’s special, I think) played Susan, the Doctor’s granddaughter, in the very first serial. (Arguably the ~first companion). There was a video (not BBC produced) in 1994 called Shakedown: Return of the Sontarans, in which she played a character called Zanelle. And then in the Big Finish audio production Forever, she voiced Unit 190.
Jacqueline Hill played Barbara, one of Susan's school teachers (also in the very first serial), who winds up travelling in the TARDIS by mistake while meeting her guardian. Years later, she also played Lexa in Meglos.
Karen Gillan played a Soothsayer in The Fires of Pompeii, and of course was then Amy Pond, the Doctor’s companion, from The Eleventh Hour to The Angels Take Manhattan.
Freema Agyeman played Adeola, a Torchwood employee, in Army of Ghosts. She also played Martha Jones, the companion in the third series of the revival. There was a brief comment about Martha’s cousin working at the former Torchwood building in one of her early episodes.
Bernard Cribbins played Donna’s grandfather, Wilf, towards the end of the Tenth Doctor’s reign. A little dubious, but he also appears as Tom Campbell in the movie Daleks – Invasion Earth, almost forty-five years earlier. And he provided the voice for Arnold Korns in Horror of Glam Rock. (Cribbins and Barnaby Edwards, who directed Glam Rock, believe it was this role which led him to be offered the role of Wilfred Mott.)
Nicholas Courtney is most famously known as Brigadier Alistair Gordon Lethbridge-Stewart (but usually just called the Brigadier), who appeared in the show regularly during the third Doctor's tenure, but also appeared occasionally with the second, fourth, and fifth Doctors. Prior to this, he appeared in The Daleks’ Master Plan as a soldier named Bret Vyon.
John Levene was another UNIT stalwart, famously known as Sergeant Benton. He also appeared as a Cyberman in The Moonbase, then as a Yeti in The Web of Fear.
Eve Myles played Gwyneth in The Unquiet Dead and Gwen Cooper in Torchwood. This was acknowledged in Journey’s End by the Tenth Doctor, with a comment about “spatial genetic multiplicity” – that is, a familial connection between them.
Lalla Ward played Princess Astra of Atrios in The Armageddon Factor, and then the second regeneration of Romana, one of the Fourth Doctor’s companions. When she regenerated, the Fourth Doctor said something along the lines of, “You can’t look like that” to which she replied, “We’re not going back to Atrios, are we?” Thanks to Amanda for pointing this out in the comments.
Ian Marter played Lieutenant John Andrews in "Carnival of Monsters", a Third Doctor story. He returned as Harry Sullivan, a companion of the Fourth Doctor, from "Robot" to "Terror of the Zygons", and then guest-starred (after being written out as a companion) in "The Android Invasion"
Minor secondary roles
Reg Cranfield is believed to have played the policeman in the opening scenes of The Pilot Episode and The Unearthly Child, which would make him the very first actor to appear on Doctor Who. (Some people dispute the casting). He then went on to make eight more appearances. (A full list is on the Doctor Who Wiki).
Bernard Horsfall played a ~lot of characters. He played the Time Lord Chancellor Goth in The Deadly Assassin. He also played a Time Lord served on the tribunal which exiled the Second Doctor in The War Games. In The Mind Robber, he played Lemuel Gulliver (a fictional adventurer and the protagonist of the book Gulliver’s Travels). In Planet of the Daleks, he was a Thal called Taron. There’s also speculation that he was the Time Lord presiding over the Doctor’s trial in The War Games was Goth, although this has never been canonically verified.
Alexander Armstrong is the voice of Mr. Smith in The Sarah Jane Adventures, and also makes a brief appearance (or whatever the word is for voices) in Journey’s End. But he also played Reg Arwell in The Doctor, the Widow and the Wardrobe, a Christmas special.
Mark Gatiss according to IMDB played the aged scientist Lazarus in The Lazarus Experiment. He was the voice of Danny Boy in Victory of the Daleks and A Good Man Goes to War. He was Gantok (the chess player) in The Wedding of River Song. Most recently, he was the Captain in Twice Upon a Time. He also wrote for the show – I believe he’s the only person to have both written for and starred in Doctor Who.
Chipo Chung appeared as Chantho in Utopia, the Master’s companion. She also appeared as the fortune teller in Turn Left.
Ellen Thomas was Carmen in Planet of the Dead, and foretold the death of the Tenth Doctor. She also played a clockwork drone in The Girl in the Fireplace.
James Marcus played a peasant in the Third Doctor story Invasion of the Dinoaurs, and then a guard called Rask in the Fourth Doctor episode Underworld.
Clive Merrison played Jim Callum in The Tomb of the Cybermen and the Deputy Chief Caretaker in Paradise Towers. Although both human characters, they’re separated by about four centuries of in-universe time, so unlikely to be the same person.
Zienia Merton played a Chinese girl, Ping-Cho, in Marco Polo, one of the first episodes broadcast, and later played the Registrar in the SJA story The Wedding of Sarah Jane Smith.
I’m not sure if Paul Kasey’s face has ever appeared on screen, but he’s played a lot of monsters in the new series. More than I can count; see Doctor Who Wiki for details.
Jessica Hynes played Joan Redfern in Human Nature and The Family of Blood, and the Tenth Doctor then visits her great-granddaughter, Verity Newman, in The End of Time.
Deep Roy played the murderous cyborg Mr. Sin in The Talons of Weng-Chiang, and then a diplomatic delegate for the reptilian Posicarians in Mindwarp.
Neve McIntosh has played three Silurian characters: Alaya in The Hungry Earth, her sister Restac in Cold Blood, and the recurring Madame Vastra from A Good Man Goes to War onwards.
Dan Starkey has appeared as a number of Sontaran characters: Commander Skorr in The Poison Sky and The Sontaran Strategem, an uncredited role in The End of Time, Part 2 as the Sontaran trying to kill Martha and Mickey, the two hapless Sontaran "invaders" in The Time of the Doctor, and (most famously) the recurring Sontaran nurse/butler Strax. In a non-Sontaran appearance, he played Ian, one of the comedy Elves, in Last Christmas.