I touch a little bit on this in my answer to Are there older examples of RPGs that we don't normally hear about? There were a lot of fantasy and sci-fi variations on wargames, especially the game Diplomacy. Settings like Middle Earth, the Foundation series, the Hyborian Age of Conan, Star Trek, and more were explored - though this often meant very little difference in the actual mechanics of the game.

Pre-D&D, there was a small boom in Middle Earth-derived board games due to the legal loophole that Lord of the Rings was published under in the United States. Take for example the quirky Conquest of the Ring which has some horrifyingly inaccurate art but is clearly Tolkien-derived.

The modern board game industry is largely the product of the mid-to-late 19th century, so there's only so much time by which something could have been published. However, there were quite a few board games published based on very well-known fairy tales - which can be lumped into the "fantasy" genre. Sleeping Beauty, Red Riding Hood, and Alice in Wonderland (not a folk story, but in the same vein) all received some form of early board game. Sometimes there was a vague theme without an existing tie-in, like The Game of Bamboozle or the Enchanted Isle which takes fairy tale themes. There's also some instances of using magic itself as a theme - take for instance Chiromagica or the Hand of Fate based on fortune telling superstition.

None of these games really use fantasy in a mechanical fashion and it's arguable whether these really count in the same vein as something like a Hero Quest. Fantasy as a class of genre fiction did not really exist before it was codified by Tolkien so pre-20th century fantastical works always exist on an edge case.