Can Rescue Rabbit still use its effect if negated by Effect Veiler since its effect resolves while it's banished in Yu-Gi-Oh?

Answer 1

Effect Veiler doesn't care where the monster is when its effect resolves, only where it the effect was activated.

Cards like Exiled Force or Rescue Rabbit activate on the field, even though they send themselves to the graveyard as the cost, and even though the effect resolves while the monsters are in the graveyard. But because the effect was activated on the field, Effect Veiler would still negate it. By contrast, if Effect Veiler was used against a Giant Rat that was destroyed by battle, then the effect would not be negated. Giant Rat activates in the graveyard, not on the field.

Skill Drain's rulings should not be referenced, because Skill Drain looks at where the monster is when the effect tries to resolve, not where the effect is activated.