Francesco Dellamorte-Dellamore in Cemetery Man - this is, in fact, the whole plot of the film.
Zac Hobson in The Quiet Earth begins suffering this when he realises that he may very well be the last human being alive. After the Good Times Montage, he begins dressing in women's clothes; he fills his garden with cardboard cutouts of celebrities; he fires his shotgun wildly at televisions; he declares himself President of the World and gives his inaugural address to the cutouts. And then the power goes out, leaving Zac standing on his balcony, in total darkness and utterly insane. He gets better.
Eric in Killing Zoe wasn't exactly the sanest person to begin with, but doing copious amounts of heroin, murdering several people, and setting off some explosives cause him to become batshit insane. As further proof, one of his favorite threats towards the end are "I'll fuck your bitch up the ass and give her AIDS!"
Stéphane in The Science of Sleep always had rather odd and confusing dreams and Imagine Spots to help his cope with reality... by the end of the film he's unable to tell the difference between the two. While it doesn't go all the way to the end he holds shades of this.
Happens to Komodo in Warriors of Virtue after he kills Master Chun. Although he was pretty zany from the start, he seemed to drop a few notches after the event.
Although Trevor Reznik from The Machinist had been acting odd ever since he kills the boy, when he starts harassing and assaulting everyone around him in a paranoid conviction that they are all out to get him you can tell he has finally completely cracked.
Mike in Deep End suffers this, mostly courtesy of Susan. The entire plot of the movie is Mike becoming her Stalker with a Crush — except it's rather hard to understand why, consideringher personality — and in the end his sanity finally snaps completely and he kills her. Given how she treated him throughout the entire movie, it's hard to sympathize with her.
Gordon in Session 9, although whether it's insanity or some sort of supernatural possession is left deliberately ambiguous.
The main character in I Am Legend suffers the beginnings of sanity slippage, asking a mannequin to talk to him because he promised his dog he'd ask, then breaking down in tears because the mannequin does not answer. It's even worse in the deleted scenes/uncut version. When he drives past the mannequin trap the head of the mannequin actually moves, and when he wakes up after being saved from his suicidal assault, he first sees the people in his house as his own wife and daughter who had died years earlier.
This trope is the whole point of Black Narcissus. All of the nuns find their flaws and emotional weaknesses slowly becoming more and more exaggerated. Things are worst with Sister Ruth, none too well to begin with, who turns into a Stalker with a Crush, and winds up dying a Disney Villain Death when she attempts to murder her superior, Sister Clodagh.
Ursula of The Little Mermaid. While not a good character, she was sane and composed during most of the film. However, when she transforms into Vanessa, its implied that she lost quite a bit of sanity (to the point of becoming a borderline Ax-Crazy) when turning into her, as she talks to her mirror in a manner similar to a schizophrenic, emits a psychotic grin when throwing a pin at a mirror's head with enough velocity to knock the mirror back, and most certainly kill a person had that been a human being, not to mention her cackling.
Alex Forrest in Fatal Attraction grows increasingly unhinged the more Dan Gallagher tries to distance himself from her, until she reaches the point where she's willing to boil his daughter's bunny in the pressure cooker, kidnap her, and then try to knife his wife to death.
In The Assassination Of Richard Nixon Samuel Byck (who was a real person) experiences this. Initially he just seems like a slightly delusional loser, but eventually he plans to hijack a plane and fly it into the white house. Not to mention his intentions of killing his boss before that.
In Cross of Iron Steiner goes through this as he's recovering in hospital. Even more so in the ending when he goes into a laughing fit in the middle of battle after Captain Stransky asks him how to reload his firearm.
The documentary Touching The Void is about the two climbers Joe Simpson and Simon Yates in their attempt to climb the Siula Grande mountain. After Joe Simpson has had his rope cut, and has been days without food or water, he eventually becomes delirious; as he presumes himself to be close to death, the song 'Brown Girl in the Ring' by Boney M gets stuck in his head, which is a song that he hates.
In Werner Herzog's Heart of Glass, the death of a foreman glassmaker has deprived a 18th century glass factory of the secret recipe for red glass. The movie is driven by the factory owner, threatened by bankruptcy, progressively losing his already weak grasp of sanity.
In Mulholland Dr. all it took to send the mentally unstable protagonist over the edge was some knocking on the door.
Repulsion is about a troubled young woman stumbling deeper and deeper into madness.
In Shine, David starts off reasonably normal but as time goes on becomes increasingly manic and starts being less intelligible, peaking when he plays Rachmaninoff's Third Piano Concerto and in the aftermath.
In Thor, Loki's sanity starts to slip after he discovers his true Frost Giant heritage. By the time he is fighting Thor at the end of the film, he is on the verge of Laughing Mad and has clearly taken a dive off of the slippery slope. The slippage continues into The Avengers, where it's clear that his fall through the Bifrost has left him Axe Crazy and more than a little unhinged.
In The Gamers: Hands of Fate, Gary continuously hallucinates horrible things about Chibichan, a Pokemon-esque creature, because its creation brought about the cancellation of one of Gary's favorite shows ever, Ninja-Dragon Riders.
Elysium: Not that he was entirely sane to begin with, but Kruger becomes noticeably more unhinged following his facial reconstruction.
In X-Men: The Last Stand, this is why Xavier had neutered Jean Grey's mental powers to keep the Phoenix hidden. And once she's unleashed...