First of all, you need to know the origins of the three races — and Tolkien was to my awareness a devout Christian, so creationism themes are strong in this universe.
Elves and Man are known as the Children of Ilúvatar, the ultimate singular creator of the universe. They are directly created by Ilúvatar. Dwarves were created by the Valar Aulë, not Ilúvatar, from the earth and stone. They were then discovered by Ilúvatar, who subsequently accepted them as the adopted children of Ilúvatar.
The above origins should be kept in mind as the context of whether they can interbreed. We know that it is possible between Elf and Man (if they can find it within them to get it on with each other) — there are at least two such recorded romances in the Silmarillion, and the last resulted in two Half-Elf/Half-Man offspring: Elrond and Elros. The two were later offered a choice to live as Elf or Man; Elrond chose to be an immortal Elf while Elros accepted his inheritance of the Gift of Man. It is through Elros' descendants that we will see the royal line of Númenor, and later of Gondor and Arnor. Aragorn is born of this line (so you can actually say Elrond is his immortal great-to-the-power-of-almost-infinity granduncle). Most likely, being both direct Children of Ilúvatar leads to similar biology that allows interbreeding.
But what of Dwarves with any of the other two? It's not as intuitive since the two Children of Ilúvatar appeared to be created from outside this world, while Dwarves were made from materials found in this world — it is possible they are of different biologies. Unfortunately I have not read of any such instances of relationships to date; Elves and Dwarves hate each other too much, human men probably don't find dwarven women attractive and human women probably think human men are bigger and more to their liking. If only that Elf-Dwarf romance in the Hobbit turned out well, we could have seen for ourselves what their babies look like.