If there ever is a zombie apocalypse, a sudden influx of ghosts, or a resurrection of Dracula, most people will know what to do. The solution to any given monster attack has been carefully disseminated to everyone with a television over the past few decades. While some weaknesses are esoteric, the oldest option in the book is to simply Burn the Undead.
The history of using fire as a weapon is messy and often chaotic. Ancient armies frequently torched the crops of their enemies, burning objects were hurled by catapults, and entire ships were set ablaze as a form of attack. Fire is a great problem solver, whether the enemy is an army of soldiers or a swarm of zombies.
When dealing with a danger who has shamed the grave, typical violence tends to fall flat. Bodily dismemberment may slow down a zombie, and a bullet to the head may even put one down, but the shambling hoard needs something bigger. Fire is a weakness for almost every form of undead antagonist. The explanation for fire as a force that can kill the dead varies. Some enemies can continue to be threatening regardless of how much of their body is blown off. Burning them to ash solves that problem. Some monsters can heal as quickly as they can be damaged. Fire consumes whatever feeds it at a pretty impressive rate. If an enemy has a weak point, fire will find it sooner or later. However an undead monster handles itself, fire will eventually solve the problem.
The Burn the Undead trope is about as old as undead as a concept. The old folkloric tales about vampires have an endless variety of rules and restrictions, but many examples suggested fire as a method of eliminating them. This primarily seems to come from Slavic traditions, in which burning a vampire was the only way to kill them while they were awake. Many of these traditions went on to inspire the Gothic literature that would formalize the cultural image of the vampire. Joseph Sheridan Le Fanu's Carmilla was one of the first examples, but Bram Stoker's Dracula remains the most iconic. Both featured fire as the final answer to their vampire problem. The book-approved way to kill Count Dracula is a multi-stage process, but the last step is always a roaring fire. The heroes may have staked Dracula through the heart and hacked off his head, but since his body remained unburned, he's free to return again.
Just like vampires, zombies have a long history with fire. Zombies come from Haitian folklore, which didn't feel the need to mention a method of killing them. It was believed that Baron Samedi would come to bring the poor souls to a version of Heaven, but they could also be saved by feeding them salt. The work that made zombies iconic in most of the world was George Romero's Night of the Living Dead. That film did introduce the idea of burning zombies. Setting a big pile of the undead on fire is the only way to put them down for good, but it's also just a strong logistical solution for a zombie plague. Later works of zombie fiction have quibbled with the idea of fire as a method of disposing of the undead. The 1985 parody Return of the Living Dead depicted a zombie apocalypse that partially resulted from a burned corpse. The burned entrails go on to infect a cemetery full of corpses, resulting in a much bigger problem. Max Brooks' Zombie Survival Guide playfully points out the weaknesses of the Burn the Undead strategy. Fire takes a long time to destroy its victim and grants the flailing monster a new degree of danger.
Video games tend to nod toward this trope by giving fire a bit of extra damage when facing undead foes. The Elder Scrolls series features vampires, liches, and other slavering undead, all of whom will take significant damage from fire magic. In games like Resident Evil, the introduction of an incendiary weapon is usually a gigantic new asset. A grenade launcher is already a giant leap forward for a game that tends to force its heroes to count pistol bullets. Left 4 Dead occasionally grants its survivors a Molotov cocktail, which immediately handles a crowd of zombies and makes quick work of the special enemies. Everything from Legacy of Kain to Paper Mario breaks out a fire as the best weapon against undead foes.
The Burn the Undead trope is an honest evolution from the earliest days of folklore to the modern day. It's an example of new writers following the rules of old stories, consistently reintroducing them to new audiences. Even when logic is applied to the concept, fire is the most satisfying way to take care of an enemy who just won't stay dead. Whether it's a million shambling corpses or a single all-powerful vampire lord, purge them with fire.