There are half-elves, and half-orcs, but there are no half-dwarfs. God knows everyone wants to fuck elves, and even orcs can have rape-babies, and humans seem to have a habit of sticking their dick into anything and everything, so why not dwarves?
Because Tolkien. J.R.R. Tolkien is why.
In Tolkien's Lord of the Rings and The Silmarillion, the elves (and later, men) were created by CEO god Iluvatar, a god of light and love and tree-hugging who was basically a stand in for the Christian God. The dwarfs were created by Aule, a middle-management god of crafting and mining. Elves and humans can interbreed because Iluvatar re-used the elfin source code when making men. Dwarfs were built on different hardware, so they can't run the same software for making babbies.
Elrond is half-elven on his mother's side. Aragorn is 1/16th or some smaller fraction elvish. Thorin Oakenshield is a dwarf made of masterwork dwarf and dwarf from the finest craftsdwarfship, and menaces with spikes of dwarf, and don't you forget it.
Which might be why dwarves eventually go extinct...
They don't count; they're sterile, exactly like mules. At least, they do in original 2e Dark Sun; in 4e, the whole sterility thing was left unmentioned and muls got an article about being included in non-DarkSun settings, giving very strong implications that they can be a true-breeding race. Neckbeards were greatly butthurt by this, as usual, but one has to confess: this does make half-dwarves a viable, true-breeding race in their own right, ergo you can have lots of them around. And why is that a bad thing?
Dwarfs are known for their endurance against poisons, magic and illnesses. This is actually caused by their very aggressive immunological system which attacks anything not-dwarf. Had your world have medicine advanced enough, you'd learn dwarfs cannot receive a blood or organ donation from any other race, and transfusing their blood (even a small amount) to non-dwarfs triggers very strong immune response in the recipient, most of the time turning into an auto-immune response within hours.
The identification gene is recessive, which means that unless both parents have it, the gene will be dormant in their child. If a female dwarf has intercourse with member of any other race, the egg is very quickly targeted by the immune system, and has no chance of settling. Having sex with a dwarf woman on her period ends in a rather nasty rash, though some charlatans advise such acts as a cure for certain enduring STD's (it doesn't work btw, the blood merely aggravates the skin).
A male dwarf impregnating an elf, human or orc woman might very well end with an impregnated egg, but around 2nd to 3rd week the fetus starts attacking it's host, which triggers an immune response from the mother, thus terminating the pregnancy. These don't even register as a pregnancy, just a brief pregnancy scare due to late period.
There have been attempts at nurturing such conceptions, by suppressing the mother's immune system during pregnancy, but all attempts ended prematurely when the mother invariably got ill from having a bombed-out immune system. Interestingly, the half-dwarf fetus always managed to shield itself from whatever infection threatened the mother's health. Attempts at exchanging the otherwise-healthy fetus to a womb of a surrogate mother woman in similar stage of pregnancy were attempted by a lich wizard, but any detailed data was lost in the inevitable looting and sacking by heroic adventurers.
The reason for why humans, elves and orcs can interbreed but not dwarves can be simply explained. Humans, elves and orcs are not species but rather races or "breeds", while dwarves are their own species.
Consider dogs, our trusty canis lupus familiaris. A pit bull and a great Dane can have puppies with each other because, although there physical characteristics are significantly altered, their overall genetic code is similar enough to allow reproduction hence we can get things like Labradoodle. It also means that breeds of dogs and cats also have a much higher tendencies for genetic defects as to get the traits we want we often breed dogs with their siblings to exaggerate what we want at the cost of recessive diseases also becoming more common.
Now lets consider D&D, with active deities loose in the world with plenty of time it very conceivable that at some point in the distant past the gods selectively started to alter traits in some single humanoid species (Let's call her Lucy for LOL's) to create the human, elf and orc races much as we altered wolves to create our dog breeds. Corellon Larethian is said to be the god of the elves and would be responsible for creating them, while Gruumsh made the orcs and the infamous battle between the two that cost Gruumsh his eye would be the result of an argument over what form the elf/orc race would take.
The observant would note that this would mean we should see Orc/Elf hybrids in the game, however consider the size difference between and Orc and an Elf. Why it's true that you can breed a Chihuahua and a great dane, the Resulting puppy causes significant problems for the mother. So an Orc male, female Elf would need a C section, at best, in order for the baby to be born at all. Orc Female male elf on the other hand is distinctly more viable but, and lets be blunt, this situation is much less likely then an Orc male elf female combination because Orcs (being mostly evil) are more likely to rape as part of pillaging while massive culture differences keep the two apart in most other cases. On the subject of alignment as anyone who has owned dogs knows is that dog breeds have differing temperaments and attitudes explaining the fixed good and evil alignment of most races. That humans are far more Alignment flexible suggests that they are the "base" species and the gods created the Elf and Orc races from humanity leaving the unaltered humans to develop on their own.
However, remember that there are dozens of human ancestors any that while close to human, are distinct species and would run into the Mul situation mentioned above of any half breed being sterile.
So while some gods chose to use Lucy as the base line for there projects, other gods chose to use a different base structure. For the Dwarfs Moradin chose to start with Neanderthals removing traits he disliked and adding traits he did (you could almost say he forged them), Yondalla created Halflings likely from Homo Floresiensis the Human "Hobbit". The hobbits would likely also be a good starting point for the Gnomes, however no Gnome Halfing hybrids. Another good candidate for who Garl Glittergold used to create the Gonme race would be the Denisovan which were discovered in a cave in 2012 and in the 3.5 hand book Garl is not said to have created the Gnomes, but having discovered them. Now, admitly we find a lot of Homonids in caves but what the Denisova have going for them is that they coexisted with Humans and Neanderthals making them a likelier candidates then other homonids.
All three races, Gnome, Dwarf and Halfling all lack hybrids and they all have deities who claim a specific role in there creation thus forming a reason why we don't get Dwarf, Gnome and Halfling hybrids, because they are different species of hominid, rather then being a different breed.
Way back in 2e, the "Dwarves Deep" splatbook for Forgotten Realms claimed that dwarves were in fact perfectly compatible with humans, gnomes and halflings, and in fact it was necessary for them to breed outside their race, since A: only 30% of all dwarves are female, and B: 55% of all dwarves are sterile. However, such crossings always produced pureblooded (if slightly taller) dwarves.
This setting also has elf-dwarfs, but they are stated to be so super-rare that no stats have ever been given, so screw you powergamers!
In Dragonlance, dwarves not being able to crossbreed with anything is pretty much shattered. This is because of the racial myth behind dwarves: to cut a long story short; Reorx had humans who worshipped him, but he grew sick of them, so he turned them into tinker gnomes. Some of these tinker gnomes then attempted to capture a magical artifact called the Graygem of Chaos, and it turned half of them into the first dwarves and the other half into the first kender. Thusly, dwarves, gnomes and kender can all have half-human kids, because their ancestors were humans.
Also, in the same setting, dwarves and gnomes can have babies together, since dwarves were magically created from gnomes. Unfortunately, these babies are "gully dwarves", which are universally ugly, hairy, smelly retards. Dwarves and gnomes saw what they had beget with such horror that they have decreed they will never mate with each other again. Unfortunately, they didn't figure this out before there were enough gully dwarves running around to propagate their own vile selves everywhere.
In the 5e corebook, it's mentioned that many believe the Stout breed of halflings are the result of halfling/dwarf crossbreeding.
As mentioned above, Dragonlance has half-dwarves, and 3.5 made them playable in the Races of Ansalon splat-book. Half-dwarfs are rare, and almost entirely hill dwarf-blooded; the more isolationist mountain dwarves simply don't interact with humans that much, and dark dwarves pretty much never see humans, so their half-dwarves pretty much don't exist. Half-gully dwarves do NOT exist, period; no human would EVER mate with a gully dwarf.
The progeny of humans and dwarves are taller than dwarves on average, ranging in height from 5 to 5 1/2 feet. They are stockier, broader in the shoulders and hips, with a tendency towards shorter legs and wider faces when compared to humans. Half-dwarves inherit the ethnic traits of their human parent. Half-dwarves may pass as dwarves with a +8 bonus to Disguise checks, but they are usually too tall and human-like to fool their dwarven kin. Half-dwarf characters possess the following racial traits.
- Humanoid (dwarf): Half-dwarves are humanoids with the dwarf subtype.
- A half-dwarf’s base land speed is 30 feet.
- Darkvision out to 60 feet.
- Weapon Familiarity: Half-dwarves have the same weapon familiarity as dwarves.
- +2 racial bonus on Craft and Appraise checks. Half- dwarves inherit the dwarf’s keen eye for craftsmanship, and their human heritage expands this to more than just stone and metal.
- Stability: A half-dwarf gains a +2 bonus on ability checks made to resist being bull rushed or tripped when standing firmly on the ground. Though not as stocky as their dwarf parents, half-dwarves still have a lower center of gravity than humans.
- +1 racial bonus on saving throws against poison.
- +1 racial bonus on saving throws against spells and spell-like effects.
- Automatic Languages: Common, Dwarven. Bonus Languages: Any. Half-dwarves often take up their human parent’s regional language as a bonus language.
- Favored Class: Any. When determining whether a multiclass half-dwarf takes an experience penalty, his highest-level class does not count.