Do intrigue cards have to be resolved?
No, you always have to resolve the card.

The rules say that when you play an intrigue card, you follow its instructions, and none of the intrigue cards have instructions that say "if you feel like it, do X".


  Play Intrigue Card: By assigning an Agent to one of the action
  spaces on Waterdeep Harbor, you play a single Intrigue card from
  your hand. If you have no Intrigue cards, you cannot assign an
  Agent to Waterdeep Harbor.
  
  When you play an Intrigue card, you follow its instructions
  immediately. Its effect takes place just once.


(emphasis mine)

However, as GendoIkari pointed out in the comments, if he actually can't follow some part of the instructions, he would do as much as possible. He'd still be resolving the card, it's just that some instructions are impossible to follow so they don't happen. I don't think there's anything in the rules about this, but if the rules wanted you to ignore the whole card just because you couldn't do part of it, they'd presumably say so.



Side note: your instinct is right; ignoring what cards say in games is pretty much always wrong. If a game has an object (a card, tile, board space, or anything) with instructions printed on it, and you play/use that object, you follow the instructions (insofar as you can). If the game doesn't want you to always follow the instructions, it'll say so in the rules.