Making D&D biology make sense is a pretty crazy feat but it's certainly possible to try.

Halflings, Gnomes, Dwarves, Goliaths and Dragonborn are all fully distinct species, they can only breed with members of their race.

Devas don't breed.

Tieflings, being mostly human, can still mate with humans to produce children, presumably the infernal genetic change has rendered them unable to mate with the other species that normal humans can mate with. They breed true, always producing tieflings.

Shifters function the same way, they can breed with other shifters or with humans and breed true. The two shifter strains are genetically incompatible.

Elves, Eladrin, Humans and Orcs (And drow for good measure). Oh my, here's the real tangle. What's the origin of humans, eladrin and orcs?

First question, are half-orcs a result of natural breeding between humans and orcs or is it result of divine intervention? Divine intervention makes things pretty simple, they're a created species that can breed with both humans and orcs. So we'll go with the more complicated natural breeding option.

Eladrin are the progenitor race for elves and drow at the minimum, so let's posit that they're the progenitor race of humans and orcs as well. The timeline would go something like this.

-Eladrin split, drow form, elves form
-Drow become separate species from Eladrin
-Elves become separate species from Eladrin
-Elves give rise to Orcs
-Elves give rise to Humans
-Genetic drift renders Orcs and Elves unable to mate
-Humans still capable of mating with both Elves and Orcs (hopefully not at the same time, that might be awkward)

Half-Elves and Half-Orcs are the only crossbreeds. They could be sterile, it's possible, but assuming they're not.

There's three possibilities here:
One, they breed true, simple, Half-Elf offspring are always Half-Elfs.

Two, the follow some sort of very basic Mendelian genetics. Two Half-Elves have a 50% at another Half-Elf, a 25% for a Human and a 25% for an Elf. Half-Elf with a pure race would give 50% Half-Elf, 50% pure. This one seems like a lot of fun, so it's the one I'd pick.

Thirdly, the traits are graduated, a half-elf that mates with a human is more elfy then a normal human but less elfy then a normal half-elf. With he complete lack of mechanical support this one seems best avoided.

And then only one question left! Can Half-Elves mate with Half-Orcs? I would lean towards the answer being almost. Although they can get pregnant, they don't produce viable offspring. But such a neat plot hook in that almost part.