In the Forgotten Realms, drow sleep, because they can't trance: their culture and mindsets render them incapable of entering the meditative state that surface elves can, so instead they sleep 8 hours like other races.
However, they are still incapable of dreaming, as are all elves.
(How they don't all go batshit insane - by which I mean, full-on hallucinations that render them utterly incapable of any kind of interaction with reality, not simply the run-of-the-mill sociopathy that they already have - the way a lack of dreaming and REM sleep would do to a human, isn't explained. Best guess: elves aren't human, they don't function the same way humans do, they don't go insane just because they aren't dreaming)
By the way, my source on this is Elaine Cunningham's "Tangled Webs", the second of the Starlight & Shadows trilogy. The fact that Liriel Baenre sleeps (not trances) but is physiologically incapable of dreaming is actually an important plot point in the book, since it reveals that another character is lying (that other character was standing over Liriel when she woke up and said "you were having a bad dream, I was checking on you"; said character was unaware that elves don't dream).
This seems to have been forgotten/retconned, however, as in 5e's Out of the Abyss mention is made of drow trancing at several points. Personally, however, I prefer the older way, mostly because I like including things that remind us that elves, dwarves, etc., aren't human.