In D&D 5e do Dark Sun halflings still exist? Is it still part of the lore or was it scrapped with 4e?
There is no official Dark Sun content in 5E at this time.
Beyond the PHB, all materials released have been generic (most things) or focused on the Forgotten Realms (Sword Coast Adventurer's Guide). All the adventure books to date have been set in the Realms, except one rewrite of old content (Tales from The Yawning Portal) and one set in Ravenloft. None of them are easily adaptable to the blasted desert of Athas.
The upcoming Mordenkainen’s Tome of Foes is named after a Greyhawk personality, and seems to have a planar theme to it. I wouldn't hold your breath for Dark Sun content, though.
To the best of my knowledge (ie: checking the WotC site and doing some searches) I haven't heard of an official Dark Sun conversion being done by WotC.
shrugs It is possible I missed it.
Athas.org is the official site for 3.x Dark Sun and is considered first party material due to the co-ownership clause regarding content in the site's charter. They have a unofficial 5e conversion taking place over there.
There is nothing in the lore of Athas to suggest that Halflings have any reason to go anywhere. No official conversions have been provided for 5th edition, however the previous upheavals of Athas ecology have not been naturally occurring events. Given 4e seemingly chose to pick up somewhere in the middle of the Dark Sun novels (as best I could tell from the details provided) and 3 RPG books (setting, bestiary, adventure) and 3 non-sequential novels were ever published for 4th edition trying to figure out where 5th edition might pick up is a losing proposition.
That said, as of 4th edition, the halflings were still very much a viable race and if you choose to remove them you should give serious consideration as to what mechanical or thematic benefits that actually provides as well as what the cause might be.