Due to their distinctive shapes, willows often make appearances in the fantasy genre, though not all are Weeping Willows.  Some of these examples include Old Man Willow from The Lord of the Rings, The Whomping Willow and Wand woods from Harry Potter.

Old Man Willow in J.R.R. Tolkien’s The Fellowship of the Ring is an example of a more devious sort of willow.  In the chapter The Old Forest, the four Hobbits are attacked by the spirit of the willow trees while they stop to rest.  Old Man Willow attempts to smother Merry and Pippin while attempting to drown Frodo in a nearby stream, only to be stopped with the arrival of Tom Bombadil.
In Harry Potter lore, Willow wood is used as a material in wand making. 

J.K. Rowling (penning as Garrick Ollivander) states on Pottermore:
“Willow is an uncommon wand wood with healing power, and I have noted that the ideal owner for a willow wand often has some (usually unwarranted) insecurity, however well they try and hide it.  While many confident customers insist on trying a willow wand (attracted by their handsome appearance and well-founded reputation for enabling advanced, non-verbal magic) my willow wands have consistently selected those of greatest potential, rather than those who feel they have little to learn.  It has always been a proverb in my family that he who has furthest to travel will go fastest with willow” 
Notable Harry Potter characters who use willow wands include Ron Weasley and Lily Evans Potter.

The Whomping Willow in Harry Potter serves as both an entrance to a secret tunnel and as a guardian, coming to life whenever something touches it.