I'm not sure I understand
The Throne of David is symbolic and the throne of Peter is not?
Poor choice of words on my part. David's throne was a
precursor to Christ's throne---i.e., David's kingship over Israel was taken over and expanded by Christ to include the whole world. In this manner of speaking, David's throne is not symbolic, because his kingship is now held by Christ and as such has never ended. Peter's throne is a lesser throne. It is not the throne of the King, it is the throne of the Chief Admininstrator.
Will Jesus Christ take the seat of David in Jerusalem or Peter in Rome?
Lord of Lords and King of Kings---Jesus has the whole shebang, Rome, Jerusalem, Constantinople, Tokyo, Buenos Aires, Winnipeg, and Zamboanga.
If you take the literal dispensationalist view, however, both Rome and Jerusalem will be destroyed along with the rest of the earth, and be replaced with a new heavens and new earth, and the only town that I know of is the New Jerusalem and environs.
It's a great place to live, but there isn't much to see if you decide to leave town on day trips.