by fate, I don't mean what happened to them physically, the Silmarillion says that they died of starvation in the forests of Doriath. I mean their cosmological fate, what happened to their spirits. they are, as far as I know, the only half-elves who were not given an explicit choice about whether they would be considered Elves, and go to the Halls of Mandos or be allowed to sail to the Undying Lands, or whether they would be considered Men, and go to their unknown fate.

their father, Dior, was the son of Beren, a Man, and Lúthien, an Elf who explicitly chose the fate of Men, and therefore he is a Half-elf also given the fate of Men. their mother, Nimloth, was an Elf. their sister, Elwing, explicitly chose the fate of the Elves, and her descendants were also given the ability to choose

so their father has the fate of Men, and their mother has the fate of Elves, and they died without a chance to choose their fate. it is not stated that Lúthien's choice also extends to her descendants, as it does for Elwing. so there seems to be no clear answer whether they should be bound to the fate of Elves, or of Men, or allowed to choose