How do warlocks deal with the healing elixir spell from Unearthed Arcana: Starter Spells, since it's a level 1 spell and warlocks use higher-level spell slots as they level up?
Same as everyone else: they use a higher level slot, but don't gain anything beyond the 1st level effects.
When a character casts a spell, he or she expends a slot of that spell's level or higher, effectively "filling" a slot with the spell.
You can find this in the PHB, page 201, under "spell slots". Emphasis mine.