Manichaeism is the only major world religion I could think of to be completely destroyed across multiple national boundaries as the result of severe persecution. What about Manichaeism was so terrifying to every single polity in late antiquity that caused such persecution?

I'm going to restrict my answer to Iran, which is just as well, because that was after all where the prophet Mani met his fate. However, I'm sure there is much that can be added on Manichaeism in the West and in China; I just don't have much material about that on hand. My main source for this will be Manfred Hutter's chapter "Manichaeism in Iran" in The Wiley-Blackwell Guide to Zoroastrianism.

Mani was born in Mesopotamia in 216 (d. 277) of "Parthian" descent (which in this context can mean any number of things). According to the Cologne Mani Codex, he was introduced to a Judaeo-Christian sect, the Elkasaites, by his father. He was also acquaintanced with the Zoroastrian tradition, however (through his Armenian mother, according to some sources) and at age twenty-four he decided to start his own religious movement. Mani was able to gain the ear of Shapur I, who came to power in 241/242 CE, to whom he preached his ideas - the "two principles of light and darkness" and the "three times", which are both notions rooted in Zoroastrianism. Mani was able to sell this as a "reform" of Zoroastrianism, while also noting that it would be inclusive of other religious movements. The hope was that this new teaching could unite Zoroastrians, Christians and even Buddhists in Iran under a common faith. Under Shapur's thirty years of rule, therefore, Manicheaism flourished.

This changed with the ascent of Hormuzd I. Now, Hormuzd also favoured Mani, but during his range, the Zoroastrian cleric Kerdir was gaining influence, and under his successor Wahram, managed to implement his reforms to strengthen Zoroastrianism's standing in the realm against Manichaeism. Kerdir helpfully recorded his deeds in an inscription in Naqsh-e Rostam:

And the creed of Ahriman and the dews was driven out of the land and deprived of credence. And Jews and Buddhists and Brahmans and Aramaic and Greek-speaking Christianis and Baptisers and Manichaeans were assailed in our land. And images were overthrown, and the dens of demons were thus destroyed, and the places and abodes of the yazads (i.e. temples to Zoroastrian divinities) were established...

At some point, Mani faced an interrogation in the court of Wahram about his teaching. Perhaps confronted by the inquisitor Kerdir, Mani was unable to defend his teachings well enough to escape imprisonment, after which he soon succumbed. To understand Kerdir's hostility, note that he likely subscribed to an ideology along the lines of that found in the "Testament of Ardashir":

Know that kingship and religion are twin brothers, no one of which can be maintained without the other ... for religion is the foundation and kingship the pillar, and possession of the foundation has more claim than possession of the pillar.

Manichaeism was thenceforth banished to the fringes of the Sasanian realms. Now, Manichaeism had much in common with Zoroastrianism - particularly the general cosmology, but also in much of its vocabulary. How much of this was appropriation by Mani and how much was genuine influence is not entirely easy to determine. However, one clearly distinguishing aspect is that Mani has a far more negative view of humanity than Zoroastrianism generally presents. Though man has a divine soul, the creation of man was not the deed of God, but caused by the demoness Az, a primeval seductress. From this follows a generally negative view of sexuality, a consistent misogynism, and an aversion to the pleasures of the material world. To Mani, the body seems to have been a prison, more than a divinely created vessel, and the goal was liberation of man's divine spirit.

These ideas are recognizable as influences from early Christian sects as well as Buddhism, the two most prominent threats to the Zoroastrian conception of world order. But, also, by appropriating Zoroastrian terminololgy yet inverting the ethics of key values of its theology, it is no wonder that Manichaeanism was able to attract both followers (by providing something new in a familiar context), and the ire of the Zoroastrian establishment (who spent no small amount of time on combatting heresy). And what is in one context a force for internal harmony, can in another be an opening for outsiders to begin leveraging influence, and in a third a disruption of political harmony. In times and places where rulership depended so much on religious claims, ecumenism could be percieved as an attempt to undermine temporal authority.

While I can't discuss its fate in the Christian world indepth, I would argue that much the same factors as in the Zoroastrian world applies there.

Let me know if you have any followup questions!