I know this thread is old, but I can't help it, since I have returned to the LotR fandom.
I don't know if there is anything that would make interbreeding between Dwarves and Elves fail, but the way Tolkien wrote things he did seem to have the races sticking to their own kind. Half-Elves like those who gave rise to Elrond's line (Beren and Luthien, Tuor and Idril) or the Dol Amroth family are all cast as tragic. Dwarves may be longer-lived than most Men (except the most pure-blooded Dunedain, like Aragorn), but they remain mortals, like Men are.
There is also the point that Tolkien Elves, if killed, have their souls stay within Arda, only retreating to Mandos (the "House of the Dead") to rest. They have the potential, depending on how traumatic their death, and if they're conduct warrants punishment or not, of being reborn, though only one Elf ever returned to Middle Earth after dying, and Glorfindel was something of a very special case, quite possibly sent for a mission, just as the Wizards were.
The mortal races either don't go to Mandos at all, or pass through quickly on their way to somewhere else. That means that, barring the possible veracity of Durin the Deathless, they simply don't come back. That would end up a tragedy for an Elven partner. There have only been two exceptions to this - when Luthien died the second time, she was fated to follow the fate of Men and have her soul leave the World entirely, while it seems to be implied that Tuor may have been awarded Elven immortality and may still be living in Valinor.
Now on the dwarven side of the equation, Dwarves in Tolkien have a serious gender imbalance happening. Only one in three dwarves is female, and not all of them will get married as dwarves are very prone to be married to their craft. There is room to suppose a culture amongst dwarves of trying to get one of the few available dwarven women as a wife, so relationships outside their race may be less than acceptable.
What canon indicates is that the only notable friendly relationships between the two races are just that - friendships. Off the top of my head, only two come to mind, Celebrimbor is strong suggested to have been good friends with Durin III, and Legolas and Gimli. The "Telchar of Nogrod" may have also been friends with Elves when he crafted Narsil, but the only thing I can base that on is that Telchar sounds like a very elven name, and Narsil ended up in Elven hands (possibly Maglor's) before it became the sword of the king of Numenor.
So TL;DR, canon implies that Elf/Dwarf pairings never happen for issues of immortal/mortal relationships being tragic and the two races mostly stick to their own kind when looking for marriagable prospects.
As for the movie, well first off, the dwarf flirting was Kíli, the younger, dark-haired brother. Fíli is the blond one. What would a child look like? Maybe shorter and stockier than an Elf, taller and thinner than a Dwarf, and possibly subject to the Half-Elven Choice (half-elves have to decide which race they want to live, and possibly die as). Tauriel and Kíli? Dark, maybe reddish hair, probably end up being an archer.
But I personally don't ship them. Aside from knowing what is going to happen to Kíli, I had a ship for him before I even entered the cinema (and it wasn't his brother, uncle, Bilbo, or any of the rest of the Company, or any other Elf) and was hoping to find reason to ignore the the thing with Tauriel, and I found it. Though I will be curious as to how far it carries out in the third movie.
Edited by Majin_Baka, 29 June 2014 - 01:17 PM.