In Dominion, how do Duration cards interact with King's Court / Throne Room?
You get $3 now, and next turn as well.

The Seaside rules contain this blurb on page 4 that clarifies most such situations:


  If you play or modify a Duration card with another card, that other card also stays in your play area until it is no longer doing
  anything. For example if you play Throne Room on Merchant
  Ship, both cards stay in play until the Clean-up phase of your next
  turn. The Throne Room stays in play to remind you that you are
  getting the effect of Merchant Ship twice on that next turn.


There are some individual exceptions for some of the more unusual effects. Throne Room can't overcome Outpost's This can’t cause you to take more
than two consecutive turns clause, and you can't Throne Room a Tactician:


  Because you must discard at
  least one card in order to gain the bonuses from Tactician, it is
  not possible to Throne Room a Tactician to get + 10 cards, +2
  Buys, and + 2 Actions.


So, bottom line, most Duration effects are simply doubled, but any time a combination with Throne Room starts getting confusing, it's best to check the rulebook - these situations are usually explicitly addressed.