Who's Responsible for Dragons Extinction in Lord of the Rings? Not Dwarves, Apparently
Well, dragons in Tolkien's legendarium were probably doomed to go extinct sooner or later.
After all, they were living instruments of destruction created by Morgoth, and though apparently capable of reproduction, they were not much interested in it, always remaining solitary and rare – and no matter how mighty an individual dragon could be, they weren't immortal.
But curiously, none of the dragons whose deaths were described or mentioned in Tolkien's canon fell by dwarven hands, even though the dwarves were the people who suffered from their attention the most throughout the Second and Third Ages of Middle-earth.
Destruction that Smaug wrought on the dwarves of the Lonely Mountain in The Hobbit was only the final chapter in the long story of woes.
Both the dwarves and the dragons loved gold and other treasures, and the only way the dragons could obtain them was to plunder someone else, with the dwarves being the most tempting targets. Almost two centuries before the sack of the Lonely Mountain there was a war which lasted about two decades, as dragons of the northern wastes descended on the Grey Mountains, the main dwarven kingdom at that point, and devastated it, forcing survivors into exile.
Back then dwarves must have been able to fight and presumably slay dragons, given that they've managed to hold out for so long, but none of the named dragons were ever slain by them.
Great Smaug, the mightiest dragon of the Third Age, was killed by Bard the Bowman. Scatha, the only other named dragon of the age, who also plundered dwarven hoards even before the dragons scoured the Grey Mountains, was killed by Fram of the Éothéod, an ancestor of the Rohirrim.
His existence was mostly remembered because of the feud between Fram and the dwarves, who demanded return of the treasures, which Scatha took from them. Instead Fram sent them a necklace of Scatha's teeth, with the message: "Jewels such as these you will not match in your treasuries, for they are hard to come by."
Though he was right, the dwarves did not take his words with good humor and the rumor persisted in the future generations that they have killed Fram in revenge.
And of course the great named dragon of the First Age perished under circumstances which had nothing to do with any dwarf. So if that rumor was true, dwarves were clearly more of a danger to dragonslayers than to dragons.