If no damage is dealt by this card's automatic ability, the passion counters on this card will not be removed because their removal is conditional on this card's automatic ability dealing damage to its target.
If Snow White is removed from the field in response to her automatic abilities being played, LKI will see the amount of counters this card had on it before it left the field, and it will deal damage equal to 300 times the number of counters this card had on it when it left.
Ex.: "Snow White, the Valkyrie of Passion" attacks while it has one passion counter, and its controller plays the second automatic ability, followed by the first automatic ability. If "Stoning to Death" is used before either ability resolves, this card's abilities will resolve and 300 damage will be dealt to the target.