In the Clash of King Game the Stannis Baratheon card says that you may take the Iron Throne token...does the defeated player retain their position on the Iron Throne track? If not, do they switch positions, or does the new player take the first position, and everyone else shifts down?

I see it as just taking the throne to get the advantage in the next bidding war, even if you can't keep it.

You take the token. You do not switch positions.