In AKIRA, a military helicopter comes out of nowhere in an early scene and lands in the middle of the road, taking up most of the scene. The helicopter sound effect doesn't start playing until after it has been onscreen for a second or two.
Berserker and Ilya's sudden appearance in the middle of an empty street early on in Fate/stay night, Justified in that Berserker (like most Servants) has the ability to recede into spirit form. (This occurs in both the anime and the game, but is definitely more apparent in the anime)
Reversed for a fantastically horrifying scene from Higurashi no Naku Koro ni. Those who've seen episode 21 will remember Mion's desperate "gomen nasai" fading out as the camera view switches to Shion and Shion isn't even paying attention anymore.
One of the more entertaining examples comes from the climax of Jurassic Park, in which the Tyrannosaurus rex - whose footsteps have been seen to make the earth tremble - appears out of nowhere, apparently having snuck in. The scene was added late in the production (the corresponding scene in the book does not feature the T-Rex) because the producers were so impressed by the T-Rex model that they wanted to show it off more.
In Alien vs. Predator, Lex is climbing up a sheer ice cliff, when she receives a call on her cell phone and answers it. If that wasn't illogical enough, she continues climbing during the conversation, and when she arrives at the top, she discovers that the person she's talking with was standing in front of a helicopter (!) at the top of that very ice cliff.
In Star Wars: Attack Of The Clones, Padmè and the Jedi rescue squad are backed into a ring in the centre of an arena. Though surrounded only by the relative quiet of droids making clicking noises, it still takes Padmè to shout "Look!" and a shot of the sky to trigger the BRRRRRRZZZZZ of the descending clone gunships. Which is doubly surprising when one considers an arena is the shape of a giant ear.
In Cloverfield, the titular monster is somehow able to sneak up on the protagonists after the helicopter crash in the middle of a park with considerably less noise and movement than one would expect, despite the fact that he was in the middle of several buildings a few blocks away perhaps a minute before. (In strict fairness, rewatching of the key scene does reveal warning footsteps and camera-jounce vibrations, but it's still somewhat credibility-stretching how quietly the monster manages to move for this key moment.)
Perhaps it was meant to show how dazed the protagonists were at the moment. Having just been in a falling helicopter is bound to be harmful to your attentiveness.
One of the most frustratingly stupid examples out there is probably in The Day After Tomorrow, where a street full of people is unable to hear or feel a 300-foot tidal wave until it's within a hundred feet of them.
In the movie version of The Lord of the Rings: The Two Towers, Aragorn hears the Riders of Rohan and pushes Legolas and Gimli to hide behind a rock. None of the Horsemen, however, notice the three heroes despite the fact that before hiding, they were standing 20 feet ahead of them, right in front of them, on top of a hill, in broad daylight. However, once they've passed and Aragorn yells at them, then they notice.
When the band of Rohirrim rides right by the three give each other quizzical looks as if to say "huh ... they didn't notice." This was on account of the cloaks from Lothlórien, which make you difficult to see against any background - green, grey, brown.
In the original they spot the Riders from very far off already (five leagues by Legolas' estimation), consciously move down to the foot of the hill so as to not present an easily visible silhouette, sit huddled in their Elven cloaks, and have to wait for some time .
The first Final Destination is among others to use the simplest version of this trope: a bus that smashes into a lead character with a spectacular roar the instant they step off the curb, but approached in utter silence before then.
Justified in The Orphanage; although a supporting character does meet their demise at the hands of a silent bus this way, the sound for the entire scene is faded to silence, as a symbolic effect showing how focused on each other the main character and the supporting character are.
The film version of Hellboy demonstrates that this trope is senseless in any sense, as Abe Sapien's introductory scene features rotten eggs which can only be smelled while in-frame, despite having been present in the scene for several minutes.
Joked about on the commentary: "Is it too late to send this back to editing?"
Perhaps Agent Meyers just assumed Professor Bruttenholm had cut the cheese and was being polite about it until he found out where the smell was really coming from?
In The Long Kiss Goodnight, a helicopter shows up from behind a house without having made a sound until the heroes saw it.
In Terminator Salvation, there is a scene where Marcus, Kyle Reese and Star are in the base of some non-Resistance humans which is being approached by a completely silent machine hunting squad consisting of a prisoner carrier, a Hunter-Killer and a HUGE Terminator. Only seconds before they attack does Star sense their presence; the sound the Terminator makes can be heard milliseconds before he strikes.
In Goldeneye, James and Natalya are under the impression that they are alone in a field. When Jack Wade approaches and gives the reinforcements the command to come out of hiding, several men pop up out of the grass. While it may be plausible to never notice the soldiers, it's quite a stretch to not notice the multiple helicopters directly overhead, which can suddenly be heard the second they drop into view.
In The Longest Day (I think) a squad of American paratroopers are holed up in a house. Despite having somebody on watch a Tiger tank sneaks up on them. It's quite funny - every time the lookout turns around for a cigarette or quip with his mates the tank silently glides out of an alley and across the street into one on the opposite side like one of the ghosts in scooby doo.
Occurs in video games as well, usually due to technical limitations. Especially egregious with scrolling games on 8-bit consoles, such as the NES. This is due to the fact that the system simply didn't handle offscreen objects; if an object left the screen, it just disappeared. (It caused notorious respawning issues and is widely seen as one of the most artifical ways possible to increase the difficulty of a game.)
The Devilsaur in World of Warcraft is a gargantuan Bigger Than T-Rex T-Rex in Un'Goro Crater, who is still able to sneak up on most players.
Developers took attempt to avert this with the Burning Crusade starting zone Fel Reaver: it ignores normal visibility range so it can be seen miles across, its steps have audible stomps and quake the earth, it frequently gives off a foghorn... And still manages to sneak up on players.
An actual gameplay element in the Dead Space series: while you can hear necromorphs in most situations, if they're sneaking up on you (especially in a vacuum), you won't be able to hear them, and there won't be a scare chord until you actually look at them. Paranoia-inducing the first time you discover this.
In The Fairly OddParents, Timmy goes back in time to see his parents when they were his age. The two of them introduce themselves to him (unaware that he's their son), and when they say their names, a conspicuously out-of-place truck blares by, totally obscuring their identities to the audience.