In brief, a heresiarch is a term for the person who originates and/or popularizes a heresy, or a false teaching. It is not a flattering description. It is related to the terms heresy and heretic. Another term for a heresiarch is "arch-heretic".
The New Testament repeatedly warns the Church of the reality of false teachers, false apostles, false prophets, etc... and exhorts her to be diligent and on her guard against these liars:
Keep watch over yourselves and all the flock of which the Holy Spirit has made you overseers. Be shepherds of the church of God, which he bought with his own blood. I know that after I leave, savage wolves will come in among you and will not spare the flock. Even from your own number men will arise and distort the truth in order to draw away disciples after them. (Acts 20:29-30 NIV)
The Spirit clearly says that in later times some will abandon the faith and follow deceiving spirits and things taught by demons. Such teachings will come through hypocritical liars, whose consciences have been seared as with a hot iron... (1 Timothy 4:1-2 NIV)
But there were also false prophets among the people, just as there will be false teachers among you. They will secretly introduce destructive heresies, even denying the sovereign Lord who bought them—bringing swift destruction on themselves. Many will follow their depraved conduct and will bring the way of truth into disrepute. In their greed these teachers will exploit you with fabricated stories. Their condemnation has long been hanging over them, and their destruction has not been sleeping.
But there were also false prophets among the people, just as there will be false teachers among you. They will secretly introduce destructive heresies, even denying the sovereign Lord who bought them—bringing swift destruction on themselves. Many will follow their depraved conduct and will bring the way of truth into disrepute. In their greed these teachers will exploit you with fabricated stories. Their condemnation has long been hanging over them, and their destruction has not been sleeping.
These examples could be multiplied. Other scriptures also warn about the danger of false teachers and false prophets (Matthew 7:15-20; 1 John 4:1-6; Jude 4).
Interestingly, while the end result is much the same, I feel that it is necessary to distinguish between false teachers and false prophets. False teachers bring false doctrines, often fallaciously based on texts of Scripture that they twist to their own destruction (2 Peter 3:16). They attack sound doctrine and in place they pervert the truth to suit their own desires. False prophets will usually do the same but they will claim exalted experiences like prophetic utterances, dreams, visions, and miraculous phenomena.
So now that we know what we're dealing with, are there examples of heresiarchs today and throughout church history? Sadly yes. Below are but a few examples:
1. Cerinthus was a heresiarch who popularized the gnostic teaching that Jesus was a man upon whom the "Christ" spirit descended at his baptism and who promptly left him just before the crucifixion.
2. Marcion taught a similar gnostic heresy that claimed that Jesus Christ was only a phantasm who did not truly possess a physical body.
3. Arius was a bishop in the fourth century who taught that Jesus Christ was a created being and thus less than God Almighty.
4. In the modern era, Joseph Smith founded the Mormon Church. He produced false Scriptures, claimed that he had had visions of God, and that Jesus was the firstborn child of God and his wife, Satan was the second born, and everyone else on earth came later.
5. Charles Taze Russell founded the Jehovah's Witnesses
6. Gregory Boyd has been the lead advocate for the heresy of "Open Theism".
7. The late Rachel Held Evans was a champion of so-called "Progressive Christianity".
These are just a few examples. The New Testament leads us to expect that false teachers, false prophets, and false apostles will proliferate throughout the present era. The answer for Christians is to be ever diligent and ever discerning towards those who teach, prophesy, or otherwise minister in the Church. Amen.