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 pillage-babies, and humans seem to have a habit of sticking their dick into anything and everything, so why not Dwarfs?
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 Judeo-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 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.
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 Orcs, Elves and Humans can interbreed but not Dwarfs can be simply explain, Orcs, Elves and Humans are not species, but rather "breeds", while Dwarfs are not.
Consider dogs: A pit bull and a great Dane can have puppies with each other because, although there physical characteristics are significantly altered, there over all gentic code are 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 there siblings to exaggerate what we want at the cost of recessive deceases also becoming more common.
Now lets consider D&D, with active deitys 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 humonide species (Let's call her Lucy for LOL's) to create the Orc, Elf and Human 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 createing 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 there 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.