Over the course of the three Austin Powers movies, Mike Myers played Austin Powers, Dr. Evil, Fat Bastard, and Goldmember. So, in the scenes between Austin and any of the villains, Mike Myers was basically talking to, or fighting, himself.
In Resident Evil: Extinction, Doctor Isaacs clones Milla Jovovich's character Alice. While these clones do not interact for most of the movie, in the final battle sequence between Alice and Isaacs, Alice watches a clone of herself die in her arms - therefore Jovovich was watching herself die. The movie ends with Alice and a clone standing side-by-side and looking at dozens more clones. The fourth movie, Resident Evil: Afterlife, starts with Alice and her clones bringing down an Umbrella facility, resulting in two or three Alices, all played by Jovovich, featuring in shots at the same time.
Not forgetting the scene where 1985 Doc Brown has a conversation with his "1955 counterpart".
In the future cafe, time-traveling Marty and Marty Junior are both crouched down behind a bar. Marty quickly grabs Marty junior's hat off the latter's head, even though they're both played by Fox.
In Dave Kevin Kline plays both the president of the U.S. (Bill Mitchell) and the head of a local employment agency who gets hired by the Secret Service to stand in for the president (Dave Kovic). At one point President Mitchell inspects Dave to make sure he'll be convincing.
Perhaps in a nod to this, in Wild Wild West, Kline plays Artemis Gordon, who on multiple occasions impersonates President Ulysses S. Grant... also played by Kline. They interact quite a bit, with Gordon even trying to fool the villain into abducting HIM by decrying the real deal as a poor imitation. It, like the film, could have gone better.
Dr. Arliss Loveless: We'll take 'em BOTH!
Peter Sellers, after honing his gift for voices on radio (see below), became famous for this on film:
He plays three characters in The Mouse That Roared, including a woman. The film lampshades this by noting they are all descendants of the founder of their very small country.
The late Alec Guinness played eight roles in the 1949 comedy Kind Hearts and Coronets. Only two at a time ever had a conversation, though.
Kung Pow!: Enter the Fist is essentially a dubbed-over wuxia film with some new footage spliced in. Steve Oedekerk did all the dubbing himself, with a single exception (the new character "Whoa"). Essentially it's an entire movie of him talking to himself, with a single scene in which another performer is heard.
Lawrence Makoare has a short scene giving orders to himself, as he plays both the Witch King and Gothmog (the butt-ugly chief orc). The two halves of the conversation were filmed months apart.
Gimli (played by John Rhys-Davies) talking to Treebeard (voiced by John Rhys-Davies).
Andy Serkis, famous for portraying Gollum, does the voices for a number of orcs and Uruk-hai. Particularly, the argument at the beginning of Two Towers about whether they should eat the hobbits? All Andy Serkis, Talking to Himself.
"Manos" The Hands of Fate was apparently so cheaply filmed, the camera could not record sound and as such, all the voices had to be dubbed in later and were done so with just four people. As Joel pointed out in the Mystery Science Theater 3000 version, there is a scene where it is fairly obvious that one person is voicing two characters in conversation.
Moon: Other than his robot buddy and a few barely-seen side-characters, this movie is entirely Sam Rockwell and his clone, interacting and at one point even having a fistfight.
TRON has David Warner playing Senior Executive VP Dillinger, the Master Control Program, and a villainous program named Sark. Sequences in both worlds have the MCP interacting with the other two. In this case, he deliberately makes no effort to change his voice for the various characters; in Tron, programs are meant to resemble their creators and are all played by the same actors; Dillinger wrote both Sark and the MCP, so all three share the same actor. Though for the MCP, Warner's voice was modulated to a lower pitch.
Star Wars has this in the prequel films where Temuera Morrison plays Jango Fett and also provides the voices for all of the clone troopers. Justified as it is firmly established in the plot that Jango's DNA was the genetic template from which the clones were created. The clones are usually wearing full body armor, allowing them to be physically portrayed by stuntmen, though Morrison does physically portray a few of the clones himself for scenes when they are seen with their helmets removed.
In Star Wars Episode V: The Empire Strikes Back, Luke Skywalker must have heard a rebel soldier announce "The first transport is away" as he ran through the Hoth base hangar. That rebel soldier is also played by Mark Hamill.
Another weird moment from Empire: an extra falling ill meant that actor Ian Liston played both rebel pilot Wes Janson and a nameless AT-AT gunner. Or as he put it, "I play a stormtrooper, in a walker, who's trying to shoot down my own bloody plane."
In The Matrix sequels, the former Agent Smith gains the ability to copy himself. Predictably, all the copies are played by Hugo Weaving. At one point, there are over a thousand copies of Smith on-screen at the same time. The fact that Neo's fight against 200+ Smith copies in the second film ends in what is effectively a draw only reaffirms how absurdly powerful of a fighter Neo is.
In the 2005 film, King Kong, Andy Serkis plays both Kong and Lumpy, the ship's cook. There's one shot on the log sequence where Lumpy fires at Kong with a machine gun... meaning that Andy Serkis is shooting at himself.
In Youth in Revolt, Nick Twisp (Michael Cera) interacts with his alter ego Francois Dillinger (Michael Cera with a silly mustache).
In Double Impact, Jean-Claude Van Damme plays twins separated at birth who reunite to avenge the murder of their parents. At one point they fight each other.
Averted in the Italian dub of the Ocean's Eleven remake. Before the film was made, Brad Pitt and George Clooney were always dubbed by the same actor. Instead of having him voice both actors, the studio had him continue to be the Italian Brad Pitt and hired another actor to be Clooney.
Similar things happened in Film/Sleepers, as both Dustin Hoffman and Robert De Niro used to be both dubbed by Ferruccio Amendola. As a result, Amendola voiced De Niro, and Hoffman was voiced by George Lopez (who kept dubbing Dustin Hoffman after the death of Ferruccio Amendola).