Standard PHB Elves cannot grow beards (but Half-Elves can).
As the PHB and the basic rules state:
Elves have no facial and little body hair.
Half-elf men do have facial hair, and sometimes grow beards to mask their
The fact that Elves cannot grow beards is specifically mentioned and is a known fact about the species - Half-Elves sometimes do it deliberately so that they look less like elves! You would expect a Belt of Dwarvenkind to cause a Half-Elf to grow a beard but not an Elf.
You are, of course, your DM, and in your setting it might be different - you could decide that Elves can grow beards normally, or just that your special cursed version of the Belt of Dwarvenkind causes beard growth even in individuals who could not normally grow beards, such as elves or most women. Since it's intended to be a negative effect in context, it makes sense that it overrides the normal rules.
BONUS PREVIOUS EDITIONS SECTION (a.k.a. I looked it up and now the question's changed scope I don't want to to just discard half my work)
The subject of Elves and facial hair has been handled consistently since 3rd edition.
In 4e Elves sort of got split up as a race, becoming Elves or Eladrin. Eladrin, who are thematically more in line with the standard "high/moon elf" of earlier work, are described such:
Eladrin can’t grow facial hair and have little body hair.
The later description of Elves omits that specific detail:
Elves have little body hair, but they favor a wild and loose look to
However, the Wizards Presents Races and Classes preview book, which came out shortly before 4e's release and talked about the design of the edition, does state:
Elves retain several of their distinguishing characteristics from earlier editions, most notably the pointed ears and the slight tilt to the eyes. And male elves don't have facial hair.
As far as I can tell, though, no such item as the Belt of Dwarvenkind exists in 4e.
The 3.5 PHB, on page 15, says of the physical description of Elves:
Elves have no facial or body hair.
The same is not mentioned in the description of Half-Elves, which does clarify that:
... their actual skin tone, hair colour and other details vary just as human features do.
(However, the 3e Belt of Dwarvenkind doesn't mention anything about causing beard growth.)
This particular facet of Elven biology appears to be in reference to the works of Tolkien which heavily inspired D&D; as per this answer over in the Sci-Fi/Fantasy stack, the Elves of Tolkien's works were almost entirely beardless, variously described as being unable to grow beards or (perhaps due to having to justify having accidentally described certain elves as having beards) only being able to do so when extremely old.