:

Berries:

Sitrus Berry would activate at 0 HP and heal 0 HP.
Figy Berry, Wiki Berry, Mago Berry, Aguav Berry, and Iapapa Berry would activate at 0 HP and heal 0 HP.
Liechi Berry, Ganlon Berry, Salac Berry, Petaya Berry, Apicot Berry, Lansat Berry, Starf Berry, Micle Berry, or Custap Berry would activate at 0 HP and get its effect.

Items:

Life Orb would deal 0 damage on attacks.
Leftovers would heal 0 HP at the end of the turn.
Black Sludge would deal 0 damage to non-Poison Pokémon at the end of the turn and heal 0 HP to Poison Pokémon.
Sticky Barb would deal 0 damage at the end of the turn.

Weather and Status:

Hail and Sandstorm would deal 0 damage at the end of the turn.
Poison would deal 0 damage at the end of the turn.

Abilities:

Solar Power would deal 0 damage in sun at the end of the turn.
Rain Dish and Ice Body would heal 0 HP in rain or hail repsectively at the end of the turn.
Dry Skin would deal 0 damage in sun at the end of the turn and heal 0 HP in rain at the end of the turn.
Volt Absorb and Water Absorb would heal 0 HP when hit by a Water or Electric move respectively.
Regenerator would heal 0 HP on switch out.
Poison Heal would heal 0 HP at the end of the turn.
Bad Dreams would deal 0 damage to sleeping Pokémon at the end of the turn.
Gluttony would activate at 0 HP and consume a berry.
Cheek Pouch would heal 0 HP after consuming a berry.
Zen Mode, Wimp Out, Emergency Exit, Shields Down and Schooling, and Beserk would activate below 0 HP.

Move Effects:

Substitute would require 0 HP to activate.
Bind, Clamp, Fire Spin, Infestation, Magma Storm, Sand Tomb, Snap Trap, Thunder Cage, Whirlpool, Wrap, G-Max Centiferno, and G-Max Sandblast would deal 0 damage at the end of the turn.
G-Max Wildfire, G-Max Cannonade, and G-Max Vine Lash would deal damage at the end of the turn.