The states are Arkansas, Maryland, Mississippi, North Carolina, Pennsylvania, South Carolina, Tennessee and Texas.

The U.S. Constitution states in Article 6 that "no religious Test shall ever be required as a Qualification to any Office or public Trust under the United States."

In 1961, the Supreme Court ruled in Torcaso v. Watkins that a person could not be denied the office of notary public for not being a believer because it "unconstitutionally invades his freedom of belief and religion guaranteed by the First Amendment and protected by the Fourteenth Amendment from infringement by the States."

Source: (no paywall) https://www.statesman.com/story/news/politics/politifact/2021/11/10/7-states-ban-atheists-office-but-bans-unenforceable/6352254001/