The card is exiled face up.

The triggered ability will resolve before the colorless spell that caused it to trigger.

The Battle for Zendikar colorless creature cards Blight Herder and Ruin Processor each have an ability that triggers when you cast it that allows you to put a card owned by an opponent from exile into that player’s graveyard. Both that ability and Thought Harvester’s ability trigger at the same time, so you can put them on the stack in any order. If Thought Harvester’s ability resolves first (meaning it went on the stack last), the card exiled by that ability can be “processed” by the other ability.