Q: Can a person who is not constitutionally eligible for the office of president be in the line of succession?
A: Such a person can serve in an office that is in the line of succession, but he or she can't become president and would get skipped over if the presidency was vacated.
To quickly review, Article II of the Constitution specifies that:
Article II: "No person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution, shall be eligible to the Office of President; neither shall any Person be eligible to that Office who shall not have attained to the Age of thirty-five Years, and been fourteen Years a Resident within the United States."
And, as we've previously discussed, the Presidential Succession Act sets an order by which government officials would ascend to the presidency. If the vice president is unable or ineligible to serve, then the speaker of the House would become president. If both the speaker and vice president were unable to do so, then the president of the Senate pro tempore would take the top job. If all three were unable to do so, then the succession would move down the following list:
Secretary of state, secretary of the treasury, secretary of defense, attorney general, secretary of the interior, secretary of agriculture, secretary of commerce, secretary of labor, secretary of health and human services, secretary of housing and urban development, secretary of transportation, secretary of energy, secretary of education, secretary of veterans affairs, secretary of homeland security.
So all of these cabinet positions are in the presidential line of succession, and some of the office holders have been, in fact, ineligible to serve as president. President George W. Bush's secretary of labor, Elaine Chao, was born in Asia. Madeleine Albright, who served as secretary of state under President Bill Clinton was born in Czechoslovakia. Both women could not have ascended to the presidency had they been called upon to do so.
Clarification, May 27: One position in the line of succession, the vice presidency, cannot be occupied by someone who is "constitutionally ineligible to the office of President," according to the 12th Amendment. However, we previously addressed how even that constitutional rule is subject to debate.
– Justin Bank
Article II, U.S. Constitution.