I know that in Dune his throne is referred to as the Golden Lion Throne, and is the symbol of his house, Corrino, but I have never found any indication of where he rules from. Does the Emperor rule from Old Earth or something?
It depends on which Emperor you are asking about; Shadam Corrino IV (Emperor during Dune, who I'm guessing you are asking about) ruled from Kaitain. He was later exiled to Salusa Secundus. (Ironic, since it was the homeworld of House Corrino.)
As far as I can see, the first reference to it is in the appendix to Dune, where it's mentioned as part of the entry on Salusa Secundus:
SALUSA SECUNDUS: third planet of Gamma Waiping; designated Imperial Prison Planet after removal of the Royal Court to Kaitain. Salusa Secundus is homeworld of House Corrino, and the second stopping point in migrations of the Wandering Zensunni. Fremen tradition says they were slaves on S.S. for nine generations.
Paul and Leto II ruled from Arrakis.
TL;DR : House Corrino ruled the galaxy from Kaitain for the better part of 10,000 years. Paul Atreides & his heirs then ruled from Arrakis after capturing the throne.
Generally when someone thinks of The Emperor in Dune, they are referring to House Corrino, which ruled the galaxy from (approximately) 88 AG until the events of Dune in 10193 AG. Although this rule was originally located on the Corrino homeworld of Salusa Secundus, it was moved early on to the Earth-like planet of Kaitain. There it remained until the Golden Lion Throne was usurped by Duke Paul Atreides, who relocated the royal court to the planet of Arrakis.
Not much is known about the planet itself as it is barely described in any of the books. What we do know is that - despite having rings - it had a climate similar to that of Old Earth, and that for centuries it was looked upon as the shining beacon of what civilization should be. Video game & fan-art depictions of the planet have tended towards balmy weather overlooking a mountainous paradise, with the capital of Corrinth City often scattered with golden buildings.