Lots of response to my question of whether West Virginia’s 91-year-old old senator, who’s third in line for the presidency, would be able to turn down the job if, God forbid, such a situation arose. In a word, yes:
I do not believe that the VP can “turn down” being president if it comes to that. After all, he ran for the job of VP knowing full well that the most significant aspect of that position (constitutionally speaking) is to be the “president in waiting”. My guess is that a Cabinet secretary would be in a similar position as nothing in the statute provides for a person to refuse to act as President (although, nothing would stop the VP or Cabinet official from taking the oath, becoming President and immediately resigning the office – thereby passing it to the next guy).
The situation with the Speaker and President Pro Tem is slightly different though, since the Constitution prohibits a sitting member of congress from serving in the Executive Branch – thus either of those officials would have to resign their congressional position before they could become President (even acting president) – a requirement also written into the statute (the Speaker or Pres. Pro Tem only act as President “upon his resignation as” Speaker/Pres PT and Representative/Senator). So, those officials would seem to be able to “turn down” the presidency by refusing to resign their congressional offices. So long as they refuse they would be ineligible to act as President and the statute would require that you move to the next person on the list.
In fact, as another reader points out, Sec. 19(d)(1) of the Presidential Succession Act of 1947 seems to provide for this:
If, by reason of death, resignation, removal from office, inability, or failure to qualify, there is no President pro tempore to act as President under subsection (b) of this section, then the officer of the United States who is highest on the following list, and who is not under disability to discharge the powers and duties of the office of President shall act as President: Secretary of State. . . .
I’d guess that “inability, or failure to qualify” covers a lot of things, including being 91 and sickly. So we can rest easy.