Turboblaze 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, Reshiram with Turboblaze can hit a Bronzong (with Levitate) using a Ground-type move like Earth Power. However, Ground-type moves would still not affect Flying-type Pokémon.
Abilities that do not directly impact the attack itself, such as Aftermath, still activate afterward. For instance, if Reshiram uses Dragon Claw against a Pokémon with Aftermath, it would still lose HP due to Aftermath.