If the speaker of the House (or the president pro tempore of the Senate or a Cabinet member) is ineligible to be president, then the line of succession would skip to the person below him. For example, Henry Kissinger could not have succeeded to the presidency when he was secretary of state.
It was different with Kissinger and with the cleaning lady—both were foreign-born naturalized citizens so they were ineligible to be President. Bill Clinton is ineligible only because he has been elected twice before. I just reread the relevant passages in the Constitution and the law on succession to the Presidency, and it seems to me that if he were speaker and the Presidency and Vice Presidency were both vacant, he would then become Acting President rather than President.