I never knew this until today. From Wikipedia:
The constitutions of these seven US states ban atheists from holding public office:

Arkansas:

"No person who denies the being of a God shall hold any office in the civil departments of this State, nor be competent to testify as a witness in any Court."[79]

Maryland:

"That no religious test ought ever to be required as a qualification for any office of profit or trust in this State, other than a declaration of belief in the existence of God; nor shall the Legislature prescribe any other oath of office than the oath prescribed by this Constitution.”[80]

Mississippi:

"No person who denies the existence of a Supreme Being shall hold any office in this state."[81]

North Carolina:

"The following persons shall be disqualified for office: First, any person who shall deny the being of Almighty God."[82]

South Carolina:

"No person who denies the existence of a Supreme Being shall hold any office under this Constitution."[83]

Tennessee:

"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."[84]

Texas:

"No religious test shall ever be required as a qualification to any office, or public trust, in this State; nor shall any one be excluded from holding office on account of his religious sentiments, provided he acknowledge the existence of a Supreme Being."