Kyon from Suzumiya Haruhi initially thinks he's the Only Sane Man after Nagato, Mikuru and Itsuki tell him the truth about their nature as well as Haruhi's. Then, Ryoko tries to stab him, traps him in a sealed dimension, turns her arms into energy tentacles, then Nagato arrives, survives getting impaled by twenty spears, chants a couple of spells, and Ryoko dissolves. Needless to say, he's afterwards quick to believe in whatever he sees or hears from these three.
Chisame in Mahou Sensei Negima! goes from the Only Sane Man into... the only sane man aware that the universe she lives in doesn't make sense at all but whatever. She gets cool age changing pills and magic hacking powers if she plays along, right?
The marines in Aliens were pretty much uniformly contemptuous of Ripley's description of the xenomorphs, refusing to believe that they could be as effectively lethal as she was saying. After all, they were rough, tough hardcore marines. Suffice to say they found religion pretty damned quick...
Captain Lorenzo in Die Hard 2 believes John McClane that Colonel Stuart and Major Grant are working together only after McClane empties a submachine gun (the same kind used by the soldiers in their attack against the terrorists) full of blanks at him. Lorenzo then calls in the cavalry.
The page quote comes from From Dusk Till Dawn, where the fact that vampires are real comes as a shock to everyone... but they get over it quickly.
In The Sixth Sense, Doctor Malcolm Crowe believes that Cole Sear's problems stem from some sort of schizoaffective disorder... until he encounters something that turns his skepticism on its head.
In Split Second, Detective Dick Durkin refuses to believe that the Serial Killer he and Harley Stone (his partner) are tracking isn't actually a human being at all, but rather some kind of monster. And then he runs face-to-face with the thing.
Stone: Did you see him?
Durkin: That wasnae a him, that was a fucking it!
Seeing the monster also leads to Durbin's line, "We need to get bigger guns!", which under the circumstances was a very rational reaction.
Doctor Peter Silberman encountered proof that Sarah Connor wasn't crazy in Terminator 2: Judgment Day. By The Sarah Connor Chronicles, he's living in isolation, away from major cities, raising dogs and keeping a healthy sense of paranoia about killer robots disguised as human beings. He even apologizes for not believing in Terminators, not that it does him much good.
However, this isn't the case in movie continuity. He appears in Terminator 3, as Scully as ever, and ascribes his experience with Terminators to psychological trauma (before recoiling in fear at the sight of the Terminator showing up again).
Then again, the item in question was the Ark of the Covenant. Not daring to look at it when some idiot is opening it is just fear of God.
Recall that one of the instructions God gave Abraham, Lot and company as they fled Sodom before its destruction was not to look back. Lot's wife disobeyed, with fatal consequences. As an archaeologist in the 1930s, Indy would have been familiar with this; indeed, he would have had an extensive knowledge of both Jewish and Christian canon and apocryphal lore. He's probably rationalizing from all this that whatever might be going on, it's something he and Marian ought not to witness if they want to live.
In The Woman in Black, Sam Daily finally acknowledges the supernatural nature of the house and the existence of the ghost in the last act, and helps Kipps lay the ghosts to rest. Well, he tries...
In The Haunting (1963), Luke Sanderson is flippantly skeptical that anything supernatural is going on at Hill House, no matter how creeped out the rest of the cast (and the audience) is getting. But by the end of the movie, even he has given in, declaring that the house ought to be burned down and sown with salt.
In Cat People, Oliver doesn't believe in the story of cat people Irena tell him about. At the end of the movie, when he finds her body shapeshifted into a panther, he realizes how wrong he was.
A skeptical atheist walks around near Loch Ness. Suddenly, the Loch Ness monster erupts from the water and grabs him. In panic, he shouts "Oh Lord, save me!" God answers: "You dare to ask me for help, when you didn't believe in Me one minute ago?" The atheist says: "Have mercy, Lord - one minute ago I didn't believe in the Loch Ness monster either!"
Paksenarrion, from Elizabeth Moon's The Deed of Paksenarrion, gradually changes from a skeptic to a believer to a full-blown Paladin of St. Gird; the Girdish medallion Canna left to Paks when Canna died gives the first signs when it starts warning Paks of dangers.
In Man from Mundania, Grey Murphy refuses to believe in magic after being brought into Xanth, until he encounters a river flowing up a cliff and simply can't come up with a scientific explanation for it.
Word of God states that Cloudtail and Mothwing of Warrior Cats stopped being atheists after an army of dead cats invaded the clan territories in The Last Hope.
Dr. Claymore from the Percy Jackson and the Olympians short story "Son of Magic." Then again, being attacked by Lamia and befriending a son of Hecate will do that to you. Not to mention dying, meeting a goddess, then coming back to the mortal world as a familiar.
Live Action TV
In the How I Met Your Mother episode "Matchmaker" Marshall and Lily (both a bit superstitious) are convinced their apartment is infested by a mutant combination of a cockroach and a mouse (which they dub a cockamouse). Robin plays the role of skeptic until the end when she finally sees the cockamouse herself (though we never do), and it's revealed that somehow the cockamouse can fly too.
LOST's Jack after he returned to the Island. Previously he denied everything supernatural or odd, even if it happened right in front of him and other people saw it. Once he became a believer, he went to the opposite extreme-believing everything. Such as lighting dynamite right next to him on a slight chance that he can't die...
On SupernaturalAgent Henriksen spent most of his time ruthlessly pursuing Sam and Dean, whom he believed to be serial killers with psychotic delusions. Seeing a mass outbreak of demonic possession up close changed his opinion on a lot of things, and he chose to help them. Sadly he was killed in the same episode.
Clive Barker's Undying: Patrick started his career as an Occult Detective trying to "debunk folklore and mysticism." Presumably he stopped trying to disprove the supernatural at the latest by some point between obtaining the clearly magical Gel'ziabar stone and gaining a German wizard as an archrival.
Girl Genius: Carson Von Mekhan, the former seneschal of Castle Heterodyne and secret ruler of Mechanicsburg during his masters' long disappearance, was of course initially dubious of Agatha's claim of being the real Heterodyne heir. Seeing Agatha in action wore down his doubts, and they were banished completely when she made the kill-happy Castle Heterodyne back down from killing Herr Diamant for doubting that there even was an heir.
Vanamonde, his grandson and successor, was dispensed from skepticism even earlier, once he had a sip of Agatha's special coffee brew and had a coffee connoisseur nirvana moment. And then he wouldn't shut up about the "perfect coffee".
Homestuck: We see in some flashbacks that the Kids have been getting contacted by the trolls periodically throughout their lives, but they've never believed that the trolls were anything other than... well, Internet trolls. When the Sburb session starts up and they get transported into the Medium, and all the other weirdness that happens, they are suddenly much more inclined to believe what the trolls have to say.
Captain Black of Jackie Chan Adventures gets this in the Season One Finale. Shendu? In ma base, steeln d talizmans? Whoa, Nellie! I'm okay now, let's defeat magic with magic!
Happens to Eduardo in the first episode of Extreme Ghostbusters. His very first line is "Ghosts. Yeah, right." Then Slimer shows up, and by the end of the episode, Eduardo is a Ghostbuster.
My Little Pony: Friendship Is Magic: In "Feeling Pinkie Keen," Twilight Sparkle is dismissively skeptical of Pinkie Pie's "Pinkie Sense," even though Applejack (who has lived in Ponyville a lot longer that Twilight) confirms that it's real. At first she insists that Pinkie's ability to predict certain events just before they happen is nothing but coincidence. Then she tries to analyze how the Pinkie sense works, because (as Pinkie points out) Twilight can't believe something she doesn't understand. After being repeatedly smacked in the face by compelling evidence, Twilight accepts that the Pinkie sense exists, even though she can't explain it.