Searching for Clues
At some point after returning from the Rushlight Tournament you'll be pestered by Linzi, who reports a most audacious theft. Seems that some thief plundered the most important bit of furniture in the land, the very seat that supports your royal hiney! Worse yet, everybody in your realm is apparently useless, leaving you to prosecute the investigation of this dastardly deed. Fortunately, Linzi does at least have some leads for you.
Leave the throne room and find Hyland in his usual spot along the eastern side of the building. Ask him about the theft of your throne and he'll relate his account of the event... which apparently included the throne animating and running away on its own. Either whatever this thief is can animate inanimate objects, of, perhaps more likely, they can turn themselves Invisible.
Make your way southwest through your capital to the merchant stalls run by Hassuf and Verdel. Talk to Verdel and you'll find in him a fellow victim of this thief who claims that magical portals were part of the burglar's repertoire. Hassuf's information is less supernatural, relating only that much of what is stolen in this city never turns up for sale.
Next head over to the tavern and talk to Elina, who will prove to be the least useful informant yet. She only relates to you information she's overheard, which paints a general pattern of behavior, but is all second-hand information. Still, it all fits with a magically-adept thief who isn't overly hindered by physical barriers of any sort. Curiously, the thief doesn't limit their acts of theft to obvious valuables, stealing apparently random junk along with the usual shiny things people value.
While all this information may have shed light on the nature of the thefts - if not the nature of the thief itself - you've precious little to go on right now. Seems like there's naught to do but wait for the stolen items to turn up on the black market. Until then, you're going to have to make do with a stool. Oh, the indignity.
The idea that you need to wait for the throne to be put up for sale is a bit misleading. What you really need to do is wander around the overworld map until you find a specific random encounter. Eventually you should find a Goblin Merchant selling a variety of wares on the side of a road - including your throne.
Talk to the merchant and pick the option "Hey, this is my throne!", after which you can ask how the goblin came to possess your throne, the answer to which really suggests you need to hire better guards. Attacking enemies in conversation usually isn't the best answer, but in this case it'll scare the goblin into giving you back your property, as well as variety of other trinkets (most likely also stolen). Most of this treasure is just loot fit for selling, but it does include two Rings of Protection +2, for what that's worth. You can also achieve the same end by succeeding at an [Intimidate 20] check, while the [Diplomacy 18] check merely convinces the goblin to raise his prices. Speaking of which, if you decide to buy your throne back, it'll lighten your purse by 100 GP, or 200 GP if you succeeded at the aforementioned Diplomacy check.
This completes the quest Runaway Throne, and for how simple it was, the experience reward at the end is certainly welcome enough.