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:
There are some individual exceptions for some of the more unusual effects. Throne Room can't overcome Outpost's
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.