Ordinances administered by unworthy clergy are invalid, and the true church is the pure church that separated from them.
Theologica lists this as a serious error, not a heresy, because its defining claim concerns the minister's worthiness and the church's purity, not the God of the Gospel. The Donatists also treated baptism as the new birth, and that teaching is rejected wherever it appears (Baptismal Regeneration).
Where It Began
During Diocletian's persecution (303–305), some clergy handed over the Scriptures to the authorities to be burned. Afterward a party of Numidian bishops refused to accept Caecilian as bishop of Carthage, because one of his consecrators, Felix of Apthungi, was accused of being such a traditor. They consecrated a rival bishop, Majorinus, who was soon succeeded by Donatus, whose name the movement took.
The quarrel began among North Africa's bishops, and both sides sought the new Christian emperor's backing. When Constantine favored Caecilian, the Donatists appealed to him. Councils of bishops at Rome (313) and Arles (314), and finally the emperor himself, ruled against them. From then on the catholic party in Africa was the state-favored church, and the Donatists were dissenters. Constantine tried force against them from 317 to 321, and in 347 the emperor's commissioners broke up their congregations with soldiers. Donatus is reported to have asked what the emperor had to do with the church, and he died in exile.
The Donatists refused to join the state-favored catholic party, and they suffered for it. Yet they kept bishops, a mistaken view of the sacraments, and apparently infant baptism. Separatists were not perfect, and the line between Christ's free churches and the institution was sometimes muddy. Theologica places them among the free churches on the timeline without endorsing their errors.
The People Involved
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Why This Is Dangerous
It rests the validity of baptism and ordination on the minister's purity, which no believer can know for certain. That moves our confidence from Christ to a man.