As of Generation 9:
Moves that don't drain an opponent's HP
Aqua Ring
Floral Healing
G-Max Finale
Grassy Terrain
Heal Order
Heal Pulse
Healing Wish
Ingrain
Jungle Healing
Life Dew
Lunar Blessing
Lunar Dance
Milk Drink
Moonlight
Morning Sun
Pollen Puff
Present
Purify
Recover
Rest
Revival Blessing
Roost
Shore Up
Slack Off
Soft-Boiled
Strength Sap
Synthesis
Wish
Source
Source
HP draining moves
Absorb
Bitter Blade
Bouncy Bubble
Drain Punch
Draining Kiss
Dream Eater
Giga Drain
Horn Leech
Leech Life
Leech Seed
Mega Drain
Oblivion Wing
Pain Split
Parabolic Charge
Source