Depending on who you ask, the Tyranids may have inspired the Trope Namers. Their basic infantry, Termagaunts wielding symbiotic guns and bounding, clawed Hormagaunts, are pretty vulnerable to anti-personnel fire, but are dirt-cheap, come in big swarms, and when near a Synapse Creature will continue to surge forward regardless of casualties. With certain rules they can even get the Without Number upgrade, allowing the player to "recycle" dead units, or you can bring along a Tervigon to replenish losses. Either this tide of chitin will overwhelm their opponent, or it will distract them until the Carnifexes, Trygons and other living siege engines smash into the enemy line. This is taken further in the background material, where the Tyranid Hive Mind employs this tactic so often that some Tyranids are born without digestive systems, as they're only intended to fight a single battle and force the enemy to expend ammunition before the real attack.
This is a perennially-popular Ork tactic as well. The close combat skill of a basic Ork Boy coupled with their low cost and large unit size means that so long as there's a good number of Orks in that mob, it probably won't break or fall back, especially if led by a Nob. With enough Boyz on the table, the enemy won't be able to stop at least some of them from getting into assault range, and it's all downhill from there. This is also shown to be what keeps the Orks going canonically; they lack intelligence, strategy, and suitable technology compared to their foes, but an Ork Waaagh can cover a planet in an ocean of Greenskins. Additionally, any time an Ork dies, which is often, they let off spores which seep into the ground and later grow even more Orks.
Although it doesnt fit their background as a dying race, the Craftworld Aeldari can pull off this tactic. Their basic unit, the Guardian, is among the weakest in entire game, and armed with a basic weapon which more-or-less shoots small shurikens at opponents. What it lacks in stopping power and accuracy it makes up for with a ridiculous rate of fire, and a unit of several Aeldari Guardians together can completely clog up an enemy target with a single attack, showering them in projectiles.
Though the Astra Militarum is famed for its tanks, its lackluster infantry is numerous and cheap (particularly fifty-man Conscript Platoons), so mass bayonet charges are a viable strategy. The catch is that unlike some other armies in the game, Guardsmen aren't driven forward by a hive mind or mob mentality, and have poor Leadership in general - enter the Commissar, who is ready to instil discipline through summary executions. Indeed, in the background material, many Imperial commanders have built their careers on throwing Guardsmen at a problem until it went away. Colonel Chenkov once filled gaps in a hastily-constructed wall by executing entire squads of his soldiers, and took a fortress without siege support in a battle that cost him ten million men but won him a medal. Prior to his removal as a playable character, Chenkov had the "Send In the Next Wave!" special rule that let him bring in a fresh Conscript Platoon each turn.
More generally, prior to the 8th Edition rules, models with the Swarm special rule operate on this principle and were represented with bases containing multiple small figures. They were particularly vulnerable to flamethrowers, explosives and the like, but each "model" typically has three times as many Wounds as other units and tended to be Fearless, allowing them to at the very least tie up an enemy squad in close combat for a couple of turns. Some of them approached Lethal Joke Character status, like Necron Scarab Swarms that were able to dismantle enemy vehicles, or Nurglings that gained deadly poisoned attacks when special character Epidemus is around.
Although very small scale compared to nearly all other Games Workshop games, it was still possible to do this with Scavvy gangs during the 1st and 2nd Editions of Necromunda as they could field a large number of weak plague zombies. Such tactics could potentially become Game Breakers due to plague zombies being incredibly cheap, moved with ridiculous speed (2D6" per turn meaning that they had a good chance of moving faster than the 8" normal humans could run) and if they took someone down in combat there was a pretty good chance they'd turn into a zombie too, in stark contrast to how hard it is to lose a ganger normally.
The 8th Edition of the game rewrote its rules to accommodate this strategy. Previously only the first rank of a block of infantry could fight at full capacity, with the second or third ranks pitching in if they had spears or pikes, but in 8th Edition the game introduced the Horde rule, allowing units to make ranks of 10 with the first three fighting, and the first two at full capacity - or in other words three times as many models could fight as before.
Orcs and Goblins (and the White DwarfGnoblar and Snotling army lists) tend to be Zerg Rushing factions for the same reasons as their Warhammer 40,000 counterparts, namely large mobs of cheap but effective infantry.
The Skaven stand out for not only having the cheapest infantry in the game, but for uniquely being able to fire upon their own men while they're locked in close combat thanks to the Life is Cheap rule.
Vampire Counts can abused this mechanic. On top of having the second most powerful combat characters in the game (second only to Chaos Lords and Greater Daemons) they can also raise more undead troops to assist them. Unlike other instances of this ability, the Vampire Counts can raise a unit beyond it's starting limits. With The End Times supplement that brought in Nagash, this took on ridiculous levels as he can create entire regiment's worth of walking corpses on the cheap, while still having enough to buff existing troops or heal them.
The Soviet Union's primary strategy in Axis And Allies is to build half a dozen infantry or more each turn. However they are only good for defense, Germany has enough industrial capacity to do this with tanks.
This is the last resort of kobolds. As Races of the Dragon and other sourcebooks explain, kobolds know they're individually pretty weak, and are therefore Combat Pragmatists who prefer to wear down attackers through traps and ambushes. If they have no other option, or their traps have failed and their homes are threatened, the kobolds' collectivist mentality leads the men to throw themselves against the enemy by the hundreds, in order to buy time for their women and children to escape with the eggs and ensure the survival of the community.
However, it is the first resort of fiendish armies during Blood War battles, who may have invented. Typically, both demons and devils send hordes of near-mindless grunts against each other, hoping a few might get through to weaken the more capable soldiers. It's not uncommon for both sides to see almost 90% casualties, and it rarely ever accomplishes much, but then, neither has the War itself.
It's also the only resort for Modrons when they form into armies (which is rare) but it makes sense if you consider their Hive Mind mentality and how they reproduce note When a modron dies, its body turns to dust and its life force is absorbed by Primus. Then, a modron of the rank immediately below the slain one evolves to fill the slain modron's position, and then a modron below that one's position is promoted, and so on down the ladder, until a new monodrone is created from within Primus. Losing 90% of their numbers is a minor setback at best for modrons, unless Primus himself is slain, and even that they can recover from. (Not easily, but they can.)
Doug Lenat won every round of the 1981 Traveller Trillion Credit Squadron tournament with PT boats, even though most of them were destroyed each game. The next year the rules were modified to emphasize mobility, which PT boats lose when damaged - so he scuttled every boat as soon as it took damage and won again. Made curious by the fact that Lenat was not a wargamer, but a computer programmer using an AI on the rulebook.
In BattleTech, this was part of the in universe purpose of the Protomech, which was smaller and cheaper than standard Battlemechs while faster and more heavily armed than Battle Armor, allowing them to be built and fielded in large numbers. In gameplay, it's sometimes invoked by flooding the map with dozens of super-cheap infantry or light vehicles, though this is typically considered to be extremely bad sportsmanship due to the Game-Breaker nature of the tactic.
An in-universe example of the tactic failing badly are one Mercer Ravannion's attempts to employ "horde tactics" by overrunning enemy forces with swarms of 20-ton 'Mechs like Stingers and Wasps (pretty much the bottom of the weight range) around the late 3010s/early 3020s...which canonically never seem to have actually succeeded at much beyond driving up his own side's casualty count against heavier opposition.
A common beginners' strategy in Risk is to gather up as many soldiers as possible in one country and go on a warpath of Curb Stomp Battles. It is possible to overcome, but it's not easy.
In Strike Legion, this is the Imperium's tried and true tactic. With three million systems under their control, a fleet of fifty million warships, and countless trillions of bodies to throw at the target, they've been wearing down the vastly smaller but vastly more-advanced and better-trained Star Republic through raw numbers. One quote from the Empress herself casually has her order an additional million ships to the front line. At the same time, the Empress has recognized that relying purely on sheer numbers is still inefficient, and has ramped up production of more advanced ships, frames, and Super Soldiers to match the Republic in quality as well as quantity.
Even with all the powers of Caine on their side, the Vampires of The World of Darkness know if the masquerade were ever broken and humanity learns of their existence, they would be wiped out in no time.
While almost every Villain in Sentinels Of The Multiverse have minions to some degree, a few of them use them and Zerg Rush to their advantage
Omnitron, as he cannot damage the heroes without his toys, relies on getting as many Components and Drones out. His "Self-Aware Robotiscs Factory" side lets him recover a Component or Drone from his trash each turn, while "Rampaging Robot" lets him play a second card each turn. It's pretty hit or miss, as he could just get a Compnent or a One-shot, but if the heroes are lagging behind and going for him instead of the Drones, he can swarm them.
Cosmic Omnitron on the other hand is much better at Zerg Rushing: His "Sentient Dropship" side lets him play the top card of his deck every time he plays a Drone. If he gets lucky, he can swarm them with Drones.
Grand Warlord Voss is a better Zerg Rusher than Omnitron. His entire deck is filled with targets, and his whole strategy relies on overwhelming the heroes with them. Ever better, his Forced Deployment card lets him revive every Minion the heroes kill, so it the heroes get a bunch at ones, they are looking at a very painful round.
The best Zerg Rusher in Sentinels is probably The Matriarch. Every time a foul enters play from her deck, she plays the top card of her deck. Every time. One foul could easily turn into seven.
The Dreamer can also turn into a Zerg Rush once she flipped to "Roused From Slumber" side. On this side, she gets extra card plays based on how many heroes she's fighting.
In X-Wing Miniatures, one fairly common Imperial archetype is to simply buy as many TIE Fighters as you can fit in the points value of the game. TIEs are much less powerful than X-wings and other Rebellion vehicles, but they're also considerably cheaper, to the point where the starter kit can provide two TIEs and one X-Wing and produce a fair fight.
In Chess already beginners get hammered in that one shouldn't zergrush. On Master level, Aljechin's Defense can be even called an inversion: lure the opponent to overzerg his centre and invade through the holes the pawns left. That much said, there is the one or other legendary game which falls under this trope, most known Kujoth-Fashingbauer, Milwaukee 1950, with eleven consecutive pawn moves that win a piece. (Knights generally hate to be on the receiving end.) And finally, there is a chess variant "Horde" where White only plays with a lot of pawns.