According to Article VI of the US Constitution:
"…no religious Test shall ever be required as a Qualification to any Office or public Trust under the United States."
That is, except in Tennessee I guess:
Article IX: Disqualifications
§ 2. Atheists holding office
No person who denies the being of God, or a future state of rewards and punishments, shall hold any office in the civil department of this State.
Hmmm… Well, they don’t allow "Ministers of the Gospel" to hold state-level legislative seats either but that is to ensure that they are "not to be diverted from the great duties of their functions" (Article IX,§1).
Now here’s a lawsuit that should be waiting to happen.