Depends are we speaking of Warcraft or World of Warcraft? In original when Demons were Demons living in hellish dimension of Twisted Nether and had to be summoned then yes they did share the traits of elements. But they were not Fel Elementals, they simply are demons as they aren't made of pure elemental essence. Basically elementals and demons are inspired by D&D Outsiders, the beings from non material plane of existence (Demons, Devils, Fiends, Angels, Elementals, Gith, Slan etc...).
In World of Warcraft however "demons" are alien species enslaved/corrupted by Sargeras and his Dreadlords and are organic as any other being from Azeroth, just addicted to "The Fel". Elementals funny enough are native to Azeroth but were banished to the elemental plane that was made as prison by the Xel'NagaTitans.
Though in both Warcraft and WoW Demons were known to be permanently killed because Blizzard forgot what demons aren't supposed to stay dead. Except Dreadlords but they are cheaters. Then again in WoW nothing is dead which can eternally be farmed and with strange patches only death has died.