What heresies have required (since the Reformation) 'rejection' after the 'first and second admonition' (Titus 3:10)?

Men teaching heresies were the subject of the admonition to expel them if they did not heed two warnings. The expulsion/rejection would not be about minor issues. Those are addressed in the need to avoid foolish questions, genealogies, strivings about the law – they are contentious issues, yes, but heresy is the matter in question. Paul wrote, “Accept him whose faith is weak, without passing judgment on disputable matters” (Romans 14:1 NIV). Two such “disputable matters” are mentioned: What to eat (vegetables only?); Whether to hold one day special, or to treat all days alike? The Titus scripture further mentions foolish questions, genealogies and arguing about the law – avoid them.

No, it is a divisive person in the church who must be warned twice, then, failing to receive such correction, is to be rejected. Subversion will be the result if that command is not heeded by the church leaders. They must be alert to protect major doctrines, such as the Godhead, the deity of Christ, the resurrection of all the dead and the day of judgment, the gospel of Christ, what saves sinners etc – all those are massive issues that the Church, from the 1st century onward is custodian of. There were many examples of the early Church identifying and expelling those in their midst who were trying to bring in destructive teachings. They soon found out the truth of this biblical warning:

“…there shall be false teachers among you, who privily shall bring in damnable heresies, even denying the Lord that bought them, and bring upon themselves swift destruction. And many shall follow their pernicious ways; by reason of whom the way of truth shall be evil spoken of” (2 Peter 2:1-2).

Such a man, for example, was Arius, eventually expelled, but perhaps too late because variations of his heresy have continued throughout the centuries, down to this very day.

But you ask about the time from the Reformation onward. It is interesting that although the Reformers never disagreed with biblical doctrines on the Godhead, the deity of Christ, the person of the Holy Spirit, the resurrection and judgment of the dead etc, they were warned, then expelled by Catholicism. One doctrine the Reformers did publicly warn about was that of Sacerdotalism, and that was one main reason for Catholicism putting them out. Thereafter, Protestantism continued to warn about the idea of sacraments only being legitimately administered by priests (even if those priests were carnal men) and this continues today. Of note is the rise of Sacerdotalism in modern groups that have (or have been) split from Protestantism. One such group that arose in the early 1800s even claimed an angel from heaven had given special (i.e. new) revelation to its founder, and authority to consecrate men as priests, who alone could administer the way of salvation. At the time, it didn’t take too long for Protestants to object then to clearly warn Christians about this heresy. That group swiftly left the established churches, repeating their founder’s testimony that God had told him all the religions of that day were corrupt, so they must have nothing to do with them.

Another group that arose a little later became attracted to the teaching of one particular man who started off as recognisably Protestant, and refused to let his group be called by a distinctive name. But it wasn’t too long before slight deviations set in, which quickly grew into open heresy, denying the full deity of Christ (having similarities to Arius), resulting in a new gospel message (which is a false gospel – Galatians 1:8-12.) They then castigated every religion apart from themselves as being ‘Babylon the Great,’ insisting all their people had to forsake all religion to join them. Protestant groups had openly condemned them by the start of the 20th century, warning people not to listen to them.

Those are two examples of 19th century groups starting out claiming to be Christian but their leaders soon brought in heretical teachings, so that Protestantism had to warn against them. Those two different heretical groups may have had leaders and followers who were initially Protestants but they very quickly separated themselves as the Bible warned:

“They went out from us, but they were not of us: for if they had been of us, they would no doubt have continued with us: but they went out, that they might be made manifest that they were not all of us” (1 John 2:19).

Given the rise of many new denominations these days, it seems as if Protestantism has become lax in warning about the heresies that are promoted by them. Of course, it’s not ‘Politically Correct’ to openly criticise those with whom you disagree, and especially not to call anyone or any group heretical! That would bring the Thought Police down on your head like a ton of bricks! No wonder we see unbiblical groups proliferating these days. They have been given the green light by society to believe what they like, and the formerly red light flashed by Protestantism has now turned amber, but is fading fast, so that soon Protestantism will have nothing to say by warning its members, let alone daring to put false teachers out!

I answer as a Reformed Protestant.