When the Pokémon Manaphy appears from a Pokéball in Super Smash Bros. Brawl, it uses its Heart Swap move to change the control of two fighters on the field, so each player controls the other one's character for a while. This doesn't mean the affected characters should commit suicide though; they still keep their own stock and damage percentages.
In Junk Man's miniquest in Mega Man Battle Network 4, Mega Man is the victim of a rockslide of garbage. Immediately afterwards, he adopts an accent and starts acting weird. Surprise — it's Junk Man, and the quest is to help him find his body (with Mega Man in it) in the rubble.
Galaxy Angel does this to the extreme with a six way swap, the entire Angel Troupe and their commander getting switched around. No voice changes ensue, resulting in a comically Camp Gay voice for Milfie-in-Volcott's-body.
Chrono Cross does this with the Hero and Villain about halfway through the story. Unusually, the switch stays for almost the entire rest of the game, with the hero having to put up with all the things the villain would have had to - except for the fact that the villain didn't really exist.
The ero game X-Change Alternative 2 centers on this trope, a departure from the series' usual theme. Ordinary high-school 2nd year Yuki Sakura is invited to watch a demonstration of his friend's new invention, an experimental teleporter device. Naturally, the machine fails (explosively) to transport matter but succeeds in swapping his mind into the body of one of his 4 female friends. The remainder of the game is spent dealing with various romance issues and hoping that the rebuilt teleporter will actually restore things to normal after spending two weeks as each other.
Occurs in Remember11, where the two main characters (Kokoro and Yukidoh) randomly swap minds throughout most of the game.
At the end of the World Builder game Psychotic!, you use a mind transfer machine to swap bodies with The Teacher.
BlazBlue: Relius and Kokonoe inflict this on a chunk of the cast in the former's gag reel.
In Fire Emblem Awakening, this happens in Sumia and Henry's support chain: while trying to teach her some Dark Magic, Henry performs a curse with this effect. If you make them advance in support level to the point where they perform a second body switch, this will trigger their Relationship Upgrade and marriage.
In the Sega Genesis/Mega Drive version of Sparkster, one of King Gedol's attacks is a spell where he can swap bodies with the titular hero. Here, you can briefly control Sparkster in Gedol's body and try to knock out Gedol in Sparkster's body to reverse the spell.
In the Haunted Castle level of Taz in Escape from Mars for the Sega Genesis/Mega Drive, after Taz gets Gossamer to sit down in one of the two chairs of the Mad Scientist's mind-swapping machine, he can switch bodies with him. Here, you can briefly control Taz in Gossamer's body in order to destroy the Mad Scientist's laser above them. After doing so, you have to switch Taz and Gossamer's minds (or lack thereof) back to their proper bodies in order to move on to the next level.
In The King of Fighters 2000, during the Korea Team's ending, while the whole team was dodging the Zero Cannon, the blast lands directly on Chang and Choi, rendering them unconscious. Kim and Jhun try to snap them (Chang and Choi) out of it and when they (Chang and Choi) wake up, not only are their bodies switched, but so are their clothes and weapons.
This is the basis of Mental Time Travel in the Zero Escape series. Certain people are born with the psychic power to SHIFT between timelines when they're in danger and going to die in one timeline, swapping their mind with a version of themselves that haven't yet gotten into that situation or (less likely) who have been in that situation but escaped it. The Zeroes are exploiting this and creating scenarios where it's impossible for everyone to escape without using Mental Time Travel to gather clues and information.