I aggroed someone in Castle Lambent, and now the throne room door is locked. I went to the Town Hall to pay the fine, but no such option appeared. Is there any way to continue the game?

Yes, just continue playing. It opens after some (in-game) time.

If you didn't kill anyone, then there might not be a fine to pay. You can check the Awards tab of the character screens to see if you have an outstanding fine.

The throne room may be closed for several months of in-game time (I seem to recall seeing 6 months, somewhere?).

-Depending on where you are in the quest line, it may be useful just to do some other quests while waiting.
-If you don't have much in the way of active quests, then make your way through the Tunnels To Eofol. You'll need to do this eventually anyway.
-Training is another good way to spend the wait time.
-Finally, if nothing else, then find a map border that takes 7-10 days to change areas, and run back and forth across it a dozen times; your characters will become weak if you run out of food, but they won't suffer any other adverse conditions (and a single night's sleep with food or at an inn will take care of the weakness).