In battle
Generation III to V
Oblivious prevents the Pokémon from being afflicted by infatuation. The Pokémon is also unaffected by Captivate.
A Pokémon with this Ability will be cured of infatuation if it is infatuated while in battle (such as if it gains the Ability by an effect like Skill Swap, or is infatuated by a Pokémon with an Ability like Mold Breaker). If the Pokémon is holding a Mental Herb or Destiny Knot, it will be eaten or activated, respectively, before Oblivious cures the Pokémon of infatuation.
Generation VI
In addition to preventing infatuation, Oblivious now prevents the Pokémon from being afflicted by Taunt.
A Pokémon with this Ability will be cured of Taunt if it is taunted while in battle (such as if it gains the Ability by an effect like Skill Swap, or is taunted by a Pokémon with an Ability like Mold Breaker). If the Pokémon is holding a Mental Herb, it will be eaten before Oblivious cures the Pokémon of the Taunt.
Outside of battle
Oblivious has no effect outside of battle.