Starship Troopers (the film) has this as the standard tactic against the giant antagonist bugs. Let me repeat that. The human infantry try to Zerg what is basically the actual Zerg. As one might expect, this fails horribly. In the book, which averts it, if 1 human soldier dies but takes 100 bugs with him, it's still a net gain for the bugs.
In The Matrix Reloaded, a fight scene between Neo and a now replicating Agent Smith basically escalates into this. It begins with Neo surrounded by maybe half a dozen or so Smith copies, and he tosses them around like rag dolls while occasionally taking a hit or two himself. From there, Smith then calls in more clones to join the fight and gains the upper hand, then Neo tips the odds back in his favor again (by improvising a weapon in the form of a metal pole), and then Smith calls in even more clones. By the end of the fight, there's maybe a hundred Smiths crowding the courtyard, and thus Neo is overwhelmed and forced to flee.
The Matrix Revolutions: The Machines play it straight when they storm Zion with any army of Sentinels. One for each human, in fact.
The ending to the 1999 film The Thomas Crown Affair features a large number of men dressed like the subject from a René Magritte painting in order to distract the authorities.
Attacks on the seven protagonists generally took place in large numbers with each attacker cut down with one or two strikes. Kurosawa is believed to have used this technique since Kenjutsu focuses on doing maximum damage in one or two cuts, and to keep the sequences interesting whilst still observing Kenjutsu's principles — a cinematic Zerg Rush was the answer.
The peasants also used their own Zerg Rush in the second battle by only letting one or two bandits into the camp at a time and then swarming them with spears from all sides.
Romper Stomper features a gang of brawny, racist skinheads who pick on one Vietnamese immigrant too many, sending endless waves of enraged Vietnamese factory workers to overwhelm and hound them across the city.
The end of Stargate features the previously oppressed slaves of Scary Dogmatic Aliens zerg rushing their former overlords, some with nothing more than sticks (or even just their bare hands), not even slowing down when some of them get killed by the panicking aliens' weapons. The fact that they cover the entire hillside when there are only a couple dozen warriors facing them makes it quite clear that they figured out the odds.
The Lord of the Rings films feature this heavily, with massive hordes of orcs swarming much smaller human armies. Possibly subverted by Sauron's use of heavier units such as trolls alongside the orcs. Also when we first see the orc army attacking (during the War of the Last Alliance) they do so mob-handed. When Sauron next makes his play for power, his armies attack in disciplined formations. At one point, Aragorn and the Army of the Dead do the same to the orcs, overpowering them in mere seconds due to being undead and thus being invincible, and simply plowing over every single enemy in their path like a plague.
300 features the Persian army most often using this tactic against the Spartans, due to their vastly superior numbers. The best example would have to be in the very first fight, which consists entirely of countless, faceless mooks charging aimlessly at the Spartan army.
Subverted (or maybe inverted) in The Gamers: Dorkness Rising where Flynn the "how different can it be" Bard becomes a Zerg Wall of Defense (in the end, literally) while the RPG Adventure Party's mage is studying up on a spell to attack The Dragon.
Used with varying degrees of success by the Na'vi in Avatar in the last battle in an attempt to overwhelm the humans' technological superiority. The aerial component, which started with a Zerg Rush out of ambush at extreme close range, works relatively well. The ground component involved cavalry charging emplaced, Dakka-laden infantrymen from extreme long range, and works exactly as much as you think it might. On the other hand, the planetary Hive Mind's own Zerg Rush with creatures the size of tractor-trailers works out much better.
In Inception, a person's dream state is populated by people generated by the subject's subconscious (i.e. "projections" as they're called in the film.) When using the film's dream-sharing technology, someone screwing around with the dream space will eventually draw the attention of the projections to the fact that there's an "outsider" in the subject's mind, and they'll then instinctively Zerg Rush the person with intent to kill.
In Zulu, spear-wielding Zulu warriors charge straight into the British camp's guns... only to stand and chant while the British cut them down. Then they leave. "He's counting your guns, testing your firepower with the lives of his warriors". "Sixty! We got at least sixty, wouldn't you say?" "That leaves only 3,940."
This is how the title insects of Antz storm the termite colony. It partially backfires, as all the termites are killed, but the only ant to survive was the one who barely did any fighting at all.
In A Bug's Life, Hopper makes it clear to his soldiers early on that they have to keep the ants' morale low, because the ants outnumber the grasshoppers a hundred to one. A rare case in which the good guys use this tactic.
Instead of the wave tactics of Wolf 359, the Federation fleet seem to employ this on the Borg in Star Trek: First Contact, with arguably better results.
This is the Empire's main strategy. Whether they be Stormtroopers on the ground or TIE Fighters in space, the Empire's strategy chiefly relies on overwhelming numbers, such that they go out of their way to ensure very few other tactics are used (for example, TIE fightercraft are rotated between pilots after every mission to ensure they aren't customized to a pilot's personal preferences). They will often supplement these forces with superweapons such as Death Stars or Super Star Destroyers; when this occurs, the main forces are simply used as a distraction or to cover potential escape routes (such as the Battle of Endor) while the Imps line up their main weapon for the kill.
Both sides use this in the Clone War. Justified by the Separatists, since they had a limitless supply of cheap battledroids. Not so justified by the Republic, which was fighting a galactic war with fewer troops than some real third-world countries command. Worse is that the Clones are frequently shown to be most adept at maneuver warfare and combined-arms attacks, but are frequently just thrown away in infantry swarms.
The cops in Gotham take on Bane's army this way in The Dark Knight Rises, since they had no time to come up with a more sophisticated strategy. It helps that both sides can't seem to hit a damn thing with their weapons, and Batman knocks out the tanks before they charge into fisticuffs.
Whether they use the Zombie Gait or just charge blindly at their targets, the zombie in films is often one that utilizes this trope to a T. Ever since "fast" zombies in Return of the Living Dead, this is a common attack by zombies in films and video games.
In the film adaptation of World War Z, the infected do not shamble toward the survivors, or run. No, they charge as an indomitable ocean of bodies. Nearly every zombie attack scene is an unbroken ocean of running monstrosities whom will sometimes charge into each other and make makeshift walls, ramp,s and ladders simply because they pile onto one another in giant piles of fallen bodies. One scene, also seen in the trailer, is where the zombies ram into a bus. By sheer number, they knock the thing right over, then continue to charge over it in a wave of bodies.
As stated above, Night of the Living Dead and all of its successors basically focus on the idea of zombies overpowering their enemies by sheer number.
In Zombieland, towards the end, this is a sequence depicted. While Columbus goes to save Wichita and Little Rock, Tallahassee ends up taking on waves of zombies trying to smash their way into the ticket booth he's hiding in as he tries to shoot down every zombie coming his way.
In Shaun of the Dead , the Winchester Pub barely holds when the zombies push the door off the hinges by sheer force of numbers, and begin overwhelming the group as they shamble in, the group unable to keep up with the massive horde.
28 Days Later has the protagonist trying to escape this on multiple situations.
The spiders from Eight Legged Freaks just try to rush and kill all of the humans in the film with reckless abandon, even when being shot at. Justifiably, spiders are not intelligent creatures, and seem to be reacting to sound.
In Dracula Untold, Mehmet responds to the news of 1,000 of his men being slaughter by having 100,000 sent for the next attack. This almost works, so Vlad makes some reinforcements of his own to even the odds.
In The Hunger Games: Mockingjay Part 1, this is how the District 5 bombers assault the power plant. Enough of them run at the guards until they overwhelm them. If the ones carrying the bomb fall, those behind them pick it up and continue to run.