Mold Breaker allows a Pokémon's moves to ignore the effects of its target's ability, but only for abilities that alter the power, accuracy, or effects of moves.
For example, a Haxorus with Mold Breaker can hit a Bronzong (with Levitate) using a Ground-type move like Earthquake. However, Ground-type moves would still not affect Flying types.
Abilities that do not directly impact the attack itself, such as Aftermath, still activate after the move. For instance, if Dragon Claw knocks out a Pokémon with Aftermath, the Mold Breaker user would still take damage from Aftermath.