For example, suppose I play a King's Court on a Lighthouse. Do I get +$3 now and +$3 next turn? +$3 now and +$1 next turn?
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.
The answer from Pat Ludwig is correct, this is just additional clarifications of the rules regarding the tracking of actions:
Donald X writes:
If you TR or KC a duration card, leave out the TR or KC with the duration card. [It's tracking that you doubled or tripled that card]
If you TR or KC a TR-or-KC used on a single duration card (and up to 2 non-duration cards), don't leave it out. [It's not tracking anything.] [You still leave out the one that hit the duration card though.]
If you TR or KC a TR-or-KC used on two or three duration cards, leave it out. [It's tracking the extra doubling/tripling done by the latter TR or KC, which also stays out.]
Therefore I am going to go back on my ruling from yesterday for what happens if you KC a KC a KC used on 3 duration cards. The 3rd KC stays out obv. The 2nd one stays out; it's indicating that the 3rd one hits 3 things. The first KC is doing nothing and so goes.
If instead you KC a KC, and that one KC's 1) a KC for 3 duration cards, 2) a duration card, 3) a non-duration, then you would keep the first KC out, as now it's indicating that the 2nd KC got to hit 3 cards, necessary for that 4th duration card to be tripled.
Slightly complicated, but the whole thread leading up to this can be found here at BoardGameGeek.com. TR = Throne Room, KC = King´s Court.