A very common weakness shared among The Undead is fire. Chopping such creatures into little bits may not stop them, but fire generally will.
The undead's weakness to fire is often connected with the idea that Fire Purifies: life-giving fire—the opposite of the dead, cold, unholy undead—is particularly effective at sending such critters back to their graves for good. In the case of vampires, a weakness to fire may be connected to, or contain, a weakness to sunlight.
Sometimes the internal justification for this trope is that the undead regenerate, and all monsters who regenerate in the setting are vulnerable to fire, likely due to the fact that fire destroys with a completeness that little else can match.
Vampires are particularly associated with this trope, but it is certainly not limited to them.
A subtrope of Fire Purifies. Frequently overlaps Fire Keeps It Dead, which is where burning prevents creatures from coming back to life in the first place (the overlap is when killing undead with fire also stops them from respawning). Also compare Burn the Witch!, which has similar reasoning behind it.
Burt Wilson: You're absolutely certain that this is gonna get rid of everything and do the trick? I mean, nothing left? [Ernie shakes his head] Ernie Kaltenbrunner: Nothing but a little-bitty pile of ashes. Burt Wilson: We don't even want the ashes! [Ernie smiles and leans over the metal grate] Ernie Kaltenbrunner: Then I'll turn it up higher, and we'll burn up the ashes, too. [Ernie slides the zombie into the oven] Ernie Kaltenbrunner: Dust to dust.
This, however, goes awry. The soot from the burn mixes with rain, animating everyone in the local graveyard, who then go on a killing/zombifying spree. Cue the Zombie Apocalypse! Electricity would have done the job better.
In The Carpenter, it turns out damaging the house causes the ghostly and homicidal carpenter pain. So, the film ends with the protagonists burning the place down, causing the carpenter to burst into flames, and burn away to nothingness.
Many of the admittedly extremely varied world folklore about vampires feature either an aversion to fire, or immolating the vampire's remains as the final step in destroying it for good.
Older Than Radio: The vampires in numerous works of Gothic literature — including Carmilla, Varney the Vampire, and Dracula must be destroyed with fire after they're staked and decapitated. The fact that Dracula's body is not burned when he's killed in the original novel is often cited as a reason for latter-day authors to bring him Back from the Dead. Again.
In P.C. Hodgell's Chronicles of the Kencyrath, fire is the best way to kill the shapeshifting, vampiric Changers, which are hard to kill but whose blood is very flammable. It's also the best way to kill the zombielike Haunts.
Fire is the best way to take care of the wights in A Song of Ice and Fire. They can also be hacked to pieces, but fire is much easier. The only body part of a wight in the series not destroyed by fire remained animate until it rotted away.
Twilightvampires are only able to die if another vampire tears them to pieces and then all of the pieces are burned. Even if they have been Driven to Suicide, this is absolutely the only way they can die, which is why Edward Cullen asks the Volturi to do this to him in New Moon. It's worth noting that Carlisle failed to try immolation during his many attempts at killing himself after he became a vampire, despite his father being a pastor who believed in wiping out evil supernatural creatures in such a manner.
Fire is the only reliable way to kill the undead in The Witch Watch. That and just cutting their heads off and living them powerless and underground whilst still being conscious.
Discussed and averted in The Zombie Survival Guide, which advises that fire is a poor weapon to use on zombies: they aren't afraid of it and can't feel pain, it won't stop them until they're literally cooked, and in the meantime you're being attacked by a flaming zombie instead of a regular one.
Fire is one of two ways to kill a werewolf the other being silver. Werewolves are considered undead in the setting. It's also said to be effective against vampires, but they have such a grab bag of strengths and weaknesses that it's not nearly as reliable.
Zombies are very strong, immortal and able to sew themselves back together if need be. However, the older they get, the drier they get, and so they're understandably nervous around fire.
The first book in the Anita Blake series has a mention that flamethrowers work quite well on vampires.
Buffy the Vampire Slayer and its spinoff, Angel, both feature vampires who can only be killed in five ways, one of which ways is setting them on fire and burning them to dust.
Dawn: If you touch my sister, hurt her in any way... you're going to wake up on fire.
In episode 42, if you looked into the living sun its soul entered your body and you effectively became undead, leaving you with only two ways to die: the soul could burn you from the inside out, or it could eat you.
The Family of Blood's scarecrows are undead and must be burned.
In ''Torchwood: Miracle Day," nobody can die. It is impossible and so the government of Britain decide that, when the hospitals get too full, anyone who should be dead is to be incinerated. Cutting their heads off and being squashed and burnt beyond recognition and given the lethal injection didn't kill them, so turning them to ashes is apparently better. The ashes may still be living, but they're not human, so the government decides it's fine to throw them away.
In Supernatural a common solution to the Monster of the Week, if its a ghost or other formerly-alive being, is to find and burn its corpse and/or bones, which utterly destroys it on this plane.
Fire is extremely useful against most undead, who are often immune to a wide variety of attack modes.
Early editions had mummies susceptible to fire damage — justified by real-world (and completely dead) Egyptian mummies being covered in flammable resins. One of the Grimtooth's Traps books combined this with Taking You with Me, by stuffing a mummy with a nice big keg of gunpowder.
Vampires in Vampire: The Masquerade take aggravated damage (very difficult to heal) from fire as well as sunlight. These two substances are the main ways to truly kill a vampire (devouring their soul also works, but only another vampire can do it). Vampires of the Setite clan are especially weak to fire, taking double aggravated damage from it. In Werewolf: The Apocalypse the shapeshifters, which generally hate vampires, have developed various magic abilities specifically to take advantage of this weakness.
The Vampire: The Masquerade/Mage: The Ascension crossover supplement "Time of Thin Blood" saw the Technocracy respond to the rising of the Ravnos antediluvian in Bangladesh by declaring Code Ragnarok... and then beating the shit out of the ancient vampire by setting him on fire with orbital mirrors after nuking him from orbit with nukes enhanced by Awakened Science. Up to that point, everything else that various supernatural groups (i.e. Garou werewolves, Asian vampires) had thrown at Ravnos had been ineffective. So, yes, on that day, the much-maligned Technocracy saved the world. Take That, mages.
In the New World of Darkness, fire just deals lethal damage to mortals and those not vulnerable to it. Vampires and Prometheans receive aggravated damage ("OW MY VERY BEING IS RENDED") from fire, however (Vampires because they're desiccated corpses held together and made lively by magic, and Prometheans because the "Divine Fire" that gives them life overloads when exposed to fire). And the Mekhet clan vampires take double damage from fire.
Mantorok and Xel'lototh zombies in Eternal Darkness can easily be killed by fire.
Final Fantasy I lists Holy and Fire as the weaknesses of Undead types. Mages usually get a fire spell early enough to use against the undead before the White Mage gets Holy.
In Half-Life, using gas canisters or oil drums to set zombies on fire is a pretty effective way of killing them. Especially used in the Ravenholm level of Half-Life 2, along with fire traps. (Setting zombies on fire is also a convenient way to light up dark areas, if you're heartless enough to listen to the human puppets being controlled by the headcrab screaming and screaming in horrible agony...
Vampires in the Legacy of Kain series can only be killed by four things: water, impalement, sunlight, and fire. In the first Soul Reaver, the Fire Reaver can shot projectiles that one-hit kills them.
The zombies in Nox will resurrect indefinitely upon being killed, unless killed with a fire spell or a fire-enchanted weapon.
The skeletal Dry Bones in Paper Mario can only be removed from a fight using fire-based attacks, to which they are particularly vulnerable. The explanation given is that burning them means there's nothing left to be reassembled. Curiously fireballs have little effect on Dry Bones in other games of the Super Mario Bros. franchise.
This is true for the MMORPG Ragnarok Online, where fire will deal up to 200% damage against undead enemies. Fire elemental weapons and spells can also be used against ghost enemies to deal 100% damage (unlike the "normal" element, dealing only 25% or even 0% damage).
Red Dead Redemption: Undead Nightmare does this. You can even burn the coffins that are making the zombies rise up from the ground. It usually takes a hit or two from a torch to make a zombie fall.
A particular monster in Resident Evil 5 plays this absolutely straight. The only way to kill it? Incinerate it in the conveniently placed furnace. Bonus points for the window that lets you watch as the creature is immolated.
Also in 5, you can use incendiary grenades and flame ammo for a grenade launcher. While they aren't super awesome zombie killers, setting those walking sacks of walking diseased flesh who are looking at you funny on fire really helps, and if they are far enough away then they will die before they get to you.
The zombies in Skylanders: Spyro's Adventure can only be damaged by fire-based attacks. If you don't have a Fire-type Skylander, you can damage them by pushing candles into them. (In the sequel, you don't need fire, but it makes things easier.)
The best way to kill Zombies in the X-COM series are with Incendiary/Phosphorous rounds, since these will automatically kill the Chryssalids/Tentaculats inside them.
In the Shining Force games fire-based magic attacks are very effective against zombies.
Fire is generally one of the best attacks to use against undead in Battle for Wesnoth — it at the very least reliably targets one of their weaker resistances, and several undead units actually take additional damage from it.
Alluded to in the The Order of the Stick prequel, On the Origin of PCs. When Roy recruits Vaarsuvius into the order, he mentions that the elf is an expert at fire magic, which will be useful since he's expecting to fight lots of undead in their upcoming quest.