What Are Organs

Souls are probably a thing in your setting, right?
And in the form of undead, these souls can probably puppet around organ-less corpses, right?
And probably, undead don't need their hearts to beat or anything, right?
But if all a body needs to operate is a soul, what are organs good forThe way it works in my setting is that biological life is a way for souls or Wills to interact more easily with the world. Anyone with enough willpower can possess a rock or wander around as a ghost, but most people can't be bothered to do so without a handy biological body. Will is independent of Life, and the undead are really just as alive as anyone else, they're just driving a broken-down piece of junk that most people would rather just toss into a grave and head to an afterlife than deal with. Explains why they're in such foul moods most the time too and why 'killing' them is typically seen as mercy rather than murder, thought the dead may disagree.

Loss of Organ Function almost always means Death, but having one of these shut down due to a really nasty roll on a Death & Dismemberment table are totally possible.

Organ Function

1.  Brain- The brain is a subtle detector of cosmic forces, a tuning fork for the unseen and immaterial. As pretty much everything is, in fact, cosmic forces, brain damage causes wide-range sensory interference, loss of motor control, and apparent madness, while the soul itself may be unfazed. Removal of the brain leaves a body unable to differentiate between elements, law and chaos, existence and nothing, and as such typically encourages a soul to vacate a body. Sticking with it is like driving a car where the windshield and mirrors and headlights are blacked out so you have to hang your head out the driver's window at 80mph to look for road signs at night. Oh and the brain is also the throne of the soul (that's why the top of the head is called the crown, innit) so in this car metaphor your car seat is also missing. Undead tend to be grumpy because their brain-car-seat is all shriveled and gross so even if it's working, it's not pleasant.
2. Skull- If the brain is the throne of the soul, the skull is the throne room. Some say it is skull shape that determines what your children will end up as and what species you count as when it comes to wizards trying to use you for spell components and who you'll be fighting against and alongside once the Skeleton War finally comes. Creatures without skulls, like worms and oozes, are viewed with great suspicion due to their lack of declared skull-allegiance. If your skull is severely broken but your brain intact, you might end up with a chicken head or something as you heal due to altered skull shape, though this is quite rare.
3. Heart- Blood is power. The heart moves blood. The heart is Strength of the Material! With no heart/blood, bodies are like the tinman in wizard of oz, rusted immobile and in desperate need of lubrication. That's why getting stabbed through the heart usually causes souls to give up their body instantly- imagine if moving your body was like trying to manually puppet your own body while standing behind it. It's a lotta dead weight for your noodly spirit arms to lift. That's also why there's two varieties of skeleton- the slow, jerky, animatronic skeletons are spirits too stupid or weak to realize walking a body with absolutely no blood or heart is a lot of work. The skeletons that are fast, skilled, and nimble are basically just ghosts who are good at juggling and have the spiritual stamina to levitate bones manually for long periods.
   
4. Liver- Seat of rage and greed and other 'base' instincts that biological life has in addition to motives of the disembodied Wills. The redder your liver is, the more inflamed your passions are, hence 'lily-livered' people tend to be rather easy-going and some might even say wishy washy and more prone to flee from troublesome situations. Alcohol inflames and reddens the liver, and certain creatures like True Elves do not have livers at all, making them able to drink endlessly and be as exactly as calm and calculating as before.
   
5. Stomach And Viscera- Negotiators of Elemental Earth- Earth enters via eating, provides life and form to the body, then leaves in the form of all solid excretions of the body. With no stomach/viscera, the earth will refuse to move for you, and so even a shallow grave will be a fearsome prison for undead lacking these parts, and as such even halfassed burials can convince the disemboweled undead to pass on rather than linger in a prison. Also, gravity will frequently give the stomachless the cold shoulder, so you can expect anything without guts to be able to fly around, provided it still has Lungs. Holy Earth, more commonly seen as sacred salts, is not as popular as Holy Water but a flung handful of it strikes undead like a hail of bullets.
   
6. Bladder- Negotiator of Elemental Water- Water enters via drinking, provides life and change and purification, then leaves in all the liquid excretions of the body. With no bladder, one cannot intentionally swim or sink, and will be entirely at the mercy of the whims of water. This is why running water is dangerous to undead and there aren't very many undead in the ocean. Dark waters, like stagnant underground lakes and frozen glaciers, aren't as strict about the elemental contact and the undead can exist in them without great issue, and of course Holy Water burns the bladderless like acid. 
7.  Lungs- Souls are ethereal things and require good airflow, or they get stiff, calcified, and/or soggy. Once an undead loses lungs, its probably going to be stuck in that body for a while. The Lungs are also the Elemental Negotiator of Air, and as such are required to speak and breathe. Without the lungs to curry the favor of Air, the wind and its thousand disparate wills devour the body one lick at a time... That's why most undead do a lot better in sheltered tombs rather than overland journeys- the wind will literally cut them to ribbons. Holy Air, more commonly seen in the form of incense, is very unpleasant but not immediately lethal to corporeal undead and is only good to ward them off, but to immaterial undead (disease spirits being the most frequently targeted) it is like a literal wall.

Decay=Elemental Taxes- The dark elements of primordial Earth, Air, and Water compose the material body, and it is light and fire, or the writhing dark of Will that animates it. But the body has a tendency to revert to its constituent parts, falling apart in stench, slime, and bones, releasing the Will within. The Negotiator Organs replace what is lost, eventually failing as interest owed to the elements becomes too great, but if the organs are destroyed prematurely, the body quickly becomes nothing but elemental parts, difficult for a soul to channel its will through.

As such, those souls wishing to hold onto a body and reign eternal in a tomb-palace of their own design rather than a secure standard-issue Afterlife maintained by a Will greater than their own would be well-advised to keep spare organs in canopic jars filled with preservatives for inevitable transplant, or resign oneself to the sacrifice of orphan children and adventurers sent to recover said orphan children, or to abandon all the pleasures of matter and exist as a phantom spirit of disembodied will, OR to turn to rather esoteric sorcerous means to replace organ function and delay entropy.


Undeath=Elemental Racketeering And/Or Tax Evasion

While your Will is your own, your body was a loaner. It's not enough to simply stitch together a bunch of fresh organs or possess some worthy vessel- without negotiation with the primordial elements, your life is not legal, cosmically speaking, and you'll have to turn to crimes against nature (that to be fair, you probably aren't even aware you're committing) to keep your false-life going. Undead that devour the living, commonly called Ghouls and Vampires, consume huge quantities of flesh and blood, wasteful, gluttonous... but by recycling flesh prematurely to their elemental forms, they earn the favor of the otherwise wrathful elementals, avoid decay, and maintain their false lives. The approach favored by the very rich who can afford opulent tombs to spend their afterlives in is of mummification, where convoluted legalese in the language of light, keeping organs outside the body when not in use, and arcane powders, bindings, and so on slow the rapacious elemental collectors of decay such that a thousand years could go by with hardly any loss of mass... provided nothing goes wrong. As for the Lich, their claim to immortal presence in the material realm comes from abilities to manufacture a new bare-bones(hur hur) body for their spirit to claim and occupy soon after the destruction of the old, without interference from psychopomps eager to whisk them off to some afterlife.

Mind you, this high-concept elemental stuff is all about the physical problems with being undead. Afterlives are like gangs, and if you didn't sign up for one during life, you'll have to fend off  'recruiters' in the form of psychopomps, demons, gods, autonomous collectives of human souls, and so on to get a moments peace if you're an unaffiliated soul with no friendly tomb to skulk in. But I've digressed quite enough even for a 3 AM post.