Which Christian Churches today teach that a Christian should not pursue a highest post in the state (like president, chancellor, prime minister, etc.)
Some Baptists and most Anabaptists preach a strict separation of church and state. Historically, this has precluded many from any type of government involvement.
Mennonites and the Amish, for example, in being called out of the world, believe they should have nothing to do with elected or high office.
According to Canon Law, Catholic priests cannot hold public office.
But anyone else can and should participate to the best of their ability. Although members of religious orders would be subject to their superiors who would probably have more meaningful duties for them.