Do immoral popes break apostolic succession?

No. A pope's personal sins, even grave and public ones, do not touch the validity of the office he holds. Christ instituted that office, and it does not depend on the moral character of the man occupying it. No, immoral popes do not break apostolic succession. The office and the man are two different things, and Scripture keeps them apart from the very beginning. Start with Peter himself. He denied Christ three times, under oath, on the night of the Passion. He was not deposed. He was restored and told to feed the sheep. Later, at Antioch, Paul withstood him to his face because he was to be blamed (Galatians 2:11). Peter had already pronounced the doctrinal decree at the Council of Jerusalem in Acts 15:7-11 that Gentiles were free of the Mosaic constraints, and then he withdrew from table fellowship out of fear of the circumcision party. That is hypocrisy, and Paul said so publicly. Notice what Paul did not say. He did not say Peter had forfeited his place. Paul rebuked him precisely because he was the leader of the Church at large; the rebuke presupposes the office rather than dissolving it. Peter accepted the correction and the office continued. Then there is Judas. In Acts 1:20 Peter applies Psalm 109:8 to him: "his bishopric let another take." The Greek is ten episkopen autou, the office itself. The traitor is removed and the office is filled by someone worthy of it. If wickedness in the holder destroyed the office, there would have been nothing left for Matthias to be elected to. Judas is the first case and the worst one, and what the apostles did about him set the pattern: replace the man, keep the office. The Old Testament works the same way. David committed adultery and then arranged the death of Uriah to cover it. He remained the anointed king. He suffered enormous temporal punishment for it, and his household was never the same, but the throne was not vacated. Nathan rebuked him and David repented; God did not owe him anything except condemnation, and mercy was given on God's initiative, not David's performance. The kingship was God's institution, not David's possession. This is why I keep telling people that infallibility is not impeccability. The gift of infallibility applies to faith and morals, to the Church teaching definitively what we must believe and do. It is not a promise that the man will be holy, prudent, or even competent. Popes make mistakes all the time. Men get promoted for the wrong reasons, in the Church as everywhere else on this earth, and the Church has had periods where the man in the chair was a scandal. When the Church invokes the charism of infallibility on a dogma, that dogma cannot go wrong. Everything outside that is the ordinary human condition, and the Church, with all the barnacles on the ship, is still the best we have on this earth. You can imagine what it is like outside her. Take Pope Liberius. He was pressured in the Arian crisis and has been accused of a lapse ever since. Yet St. Ambrose spoke of "Liberius of blessed memory," St. Basil called him "the blessed bishop Liberius," Pope Siricius called him "my predecessor of venerable memory," and the Byzantine, Coptic and Ethiopian synaxaria commemorate him as a saint. Theodoret, appealing to Pope Leo the Great, could still say of the Roman Church that it had "never been infected with any heresy." The see held even when its occupants were under fire. So on nepotism, on greed, on the Renaissance scandals: those are real sins and they deserve to be named as sins. They are an argument for reform and penance and for putting holy men in office. They are not an argument that Christ's promise failed. Succession is broken only when the Church herself lawfully judges that a man never held or has lost the office, and that judgment belongs to the Church, not to us as private individuals sitting in our living rooms deciding who is and is not pope.