This happens in Sanctuary any time Bigfoot and John Druitt/Jack the Ripper share a scene. Both characters are physically portrayed by actor Chris Heyerdahl, although when he is Bigfoot, he has quite a bit of make-up and prostheses on.
Gene Roddenberry's Andromeda: there was a three-way conversation between the ship's computer, a hologram generated by the ship's computer, and the robotic version of the ship's computer (all played by the same actress, of course).
In a segment of Beakman's World, Mark Ritts as Harry Pitts converses with Mark Ritts as Lester the Rat.
Beetleborgs did this in the episode "Buggin Out" where Flabber slowly turns into Kombat Gnat. Kombat Gnat's provided by Billy Forester who played Flabber.
Several episodes had Buffy interacting with a robotic duplicate of herself.
Season seven's Big Bad was The First Evil, who can only take the form of people who have died, which leads to Buffy and Spike both having conversations with "themselves".
Oddly enough, the season five episode where Xander is split into two people is a subversion. The actor, Nicholas Brendon, has an identical twin, Kelly Donovan. Their conversations and interactions are done completely without special effects, though through using multiple takes, and editing Nicholas Brendon still did nearly all of the dialog for both Xanders.
In the Angel episode "Orpheus", Angelus and Faith share a dream/vision where they watch Angel's tormented past. At the end, the memory of Angel becomes an active participant and physically fights Angelus (both played by David Boreanaz).
Nicholas Briggs voices both the Daleks and the Cybermen in Doctor Who, leading to a rather memorable scene in the second-series finale "Doomsday". This also means any conversation between Daleks is Briggs Talking to Himself.
In series 2 of the new Doctor Who, the Doctor, Rose, and Mickey accidentally travel to an alternate universe and meet up with Mickey's alternate self and his gang. In series 1, there was a running gag where the Doctor would call him Ricky instead of Mickey, and while the Doctor no longer gave him the name in series 2, Mickey's alternate self was called Ricky in reference to this.
In a minisode of the new Doctor Who, the Tardis materialises inside itself, meaning that when a character left the Tardis, they re-entered the Tardis. This resulted in actors Matt Smith, Karen Gillan and Arthur Darvill all interacting with themselves, including Gillan flirting with herself.
In the original Doctor Who, the 3rd Doctor (Jon Pertwee) has a brief conversation with himself (his attempt to get his TARDIS working causes his and his companion's future selves to appear for a minute). Tom Baker had conversations with himself in "The Android Invasion" and "Meglos" (The Doctor and an evil doppelganger, in both). The first Romana had a short conversation with her Identical Stranger. The 5th Doctor had a conversation with Omega, who had copied his body, but it may not count, as it was Peter Davison's body with Omega's previous voice.
Another episode features a doppleganger version of the Eleventh Doctor with the two Doctors sharing scenes together.
In The Sarah Connor Chronicles episode "Allison From Palmdale," Cameron has several lengthy and creepy conversations with Allison, the girl who her physical appearance was copied from, and at the end breaks Allison's neck. Naturally, both of them are played by Summer Glau.
Chuck (ventriloquist) and Bob (dummy) regularly had conversations with each other on Soap. (Also see below)
Michael Shanks plays both Daniel Jackson and the Asgard character Thor. (Asgard look so different from humans that they get only voice actors, although this is a live-action show). Teryl Rothery, who plays Janet Fraiser, also does the voice of an Asgard named Heindall, but these two characters never interact or even meet.
Another Michael Shanks example — in the episode "Holiday", he plays both Daniel and an alien character, Ma'chello, under heavy makeup. Ma'chello uses a machine to swap bodies with Daniel, leading to an interesting scene where Michael Shanks, as Ma'chello is Daniel's body, argues with Michael Shanks, as Daniel in Ma'chello's body!
In one episode of Mork and Mindy, Mork met Robin Williams and discussed clothing tastes.
In-universe example: On one episode of Remember WENN, after Jeff Singer leaves and before Scott Sherwood is hired as an actor, Mackie Bloom is forced to voice every single character himself, until he forgets what his own voice sounds like.
Subverted in The 7:30 Report, an Australian current affairs programme that has a weekly political satire sketch of a fake current affairs interview, starring Bryan Dawe (usually the interviewer) and John Clarke (the interviewee — a different character each episode but usually portraying an actual person. Often it's a politician). On rare occasions there will be multiple interviewees, with John playing each one. The most confusing aspect of it is that John makes no attempt to imitate who he's impersonating (i.e. they all look and sound like John Clarke in real life, as well as having a habit for dodging Bryan's questions) so at times it looks as though he really is talking to himself.
On Mystery Science Theater 3000, at least one of the "Mads" played one of the 'bots at all times. (Dr. Forrester/Crow, Dr. Erhardt/Tom Servo, Brain Guy/Crow, Bobo/Tom Servo)
When Bizarro appeared, Tom Welling was both the hero and villain.
That wasn't the first time that Tom Welling fought himself on the show. He did so as early as Season 1 in an episode when the Freak of the Week was a shapeshifter who disguised herself as Clark during her fight against the real Clark.
It also occurred when Clark's Mirror Universe counterpart, Ultraman, showed up in Season 10.
Smallville even had one Freak of the Week whose meteor ability was to literally clone himself. Predictably, the original guy and the clone appeared on-screen together being played by the same actor.
Jensen got to play two Deans in "Dream a Little Dream of Me" and "The End". The former was a demonic version of Dean, the latter Dean's future self in a Crapsack World.
Jared Padalecki plays three different versions of Sam in "The Man Who Knew Too Much", and while only two of them are ever onscreen at the same time, it still veers into this trope a couple of times.
In "Swan Song", Padalecki plays both Lucifer while he's possessing Sam's body, and Sam himself. The two hold a conversation through a mirror.
The nonexistent budget ensured that PJ Katie from PJ Katie's Farm did the voices for every character on the show.
In multiple episodes of Fringe, Anna Torv plays two version of Olivia Dunham — one from "our" universe and one from another universe. While the two characters are usually seen separately, in the season two finale "Over There", our Olivia and alternate Olivia engage in hand-on-hand combat — meaning Torv is fighting herself.
Happens in any episode of Farscape in which Crais (played by Lani Tupu) and Pilot (voiced by Lani Tupu) interact.
Peter Tuddenham did the voices of the various talking computers in Blake's 7. On one occasion Slave and Orac get into a brief tiff; Tuddenham was asked if he wanted to record one of the voices first but he did them both live.
Slave: Uh, I don't wish to interrupt, Master... Orac: Then kindly don't. Slave: I wasn't talking to you. Orac: You were attempting to override a superior system. Be silent!
Unfortunately, Slave was trying to warn the crew about an impending attack on the ship. Way to go, Orac.
While no talking was involved, an in-universe example in Ned's Declassified School Survival Guide has an instance where the gym teacher made a video on the rules of dodgeball where she played four characters.
In SCTV, there's a sketch in which Rick Moranis as Dick Cavett interviews himself. Yes, there are two Dick Cavetts on screen both played by Moranis.
On an episode of The All-New Let's Make a Deal in 1985, host Monty Hall let then-announcer Dean Goss host two deals as part of a test run. During said deals, Goss was both hosting and announcing, but according to Word of God, his announcing was pre-taped so that he could do this. (The episode was a "test run" of sorts, as Monty had planned to pass the show on for Goss to host, should it be renewed for a third season. It wasn't.)
On Orphan Black, Tatiana Maslany plays, by the end of the first season, seven different characters who spend a good deal of the show's time interacting with one another.
The Mighty Boosh: Due to the two stars playing a variety of different characters, this trope is inevitable sometimes, such as when the Hitcher interacts with Vince in the last episode of series one.