The ability holder cannot become poisoned. If the Pokémon got poisoned (either getting Poisoned before the ability was given to them or getting hit by a poison-afflicting move from a Pokémon with Mold Breaker), they get cured once their turn starts. The Pokémon consumes its Pecha Berry or other Poison-healing held item before Immunity activates.
From the fifth generation onwards as well as the third generation, if a poisoned Pokémon originally had Immunity but, currently does not, it will not heal its poison.
Outside of Battle[]
If the ability holder somehow got poisoned, they will not take poison damage. Battle Frontier events that poison won't poison this Pokémon.