Mold Breaker nullifies the following Abilities (even if they would end up boosting the power of the Mold Breaker Pokémon's move): Battle Armor, Clear Body, Damp, Dry Skin, Filter, Flash Fire, Flower Gift, Heatproof, Hyper Cutter, Immunity, Inner Focus, Insomnia, Keen Eye, Leaf Guard, Levitate, Lightningrod, Limber, Magma Armor, Marvel Scale, Motor Drive, Oblivious, Own Tempo, Sand Veil, Shell Armor, Shield Dust, Simple, Snow Cloak, Solid Rock, Soundproof, Sticky Hold, Storm Drain, Sturdy, Suction Cups, Tangled Feet, Thick Fat, Unaware, Vital Spirit, Volt Absorb, Water Absorb, Water Veil, White Smoke, Wonder Guard, Big Pecks, Contrary, Friend Guard, Heavy Metal, Light Metal, Magic Bounce, Multiscale, Sap Sipper, Telepathy, and Wonder Skin.
Mold Breaker only negates these Abilities in the event they weaken or negate moves; they are active in any other instance. While Mold Breaker allows a Pokémon to bypass the effects of moves that prevent status (i.e. Immunity, Insomnia, Limber, Own Tempo, and Vital Spirit, but not Leaf Guard), the status the Pokémon's move inflicts will be cured instantly after the attack. Lightningrod and Storm Drain still divert attacks in double or triple battles, but will not prevent attacks when they land (and will not make any Electric- or Water-type attacks hit with 100% accuracy, respectively). Dry Skin still heals or damages the Pokémon in the appropriate weather.
Those are the only abilities it activates.