The persecution's longest consequence was not the martyrs but the survivors. Clergy who had handed over the scriptures to be burned, the traditores, were widely despised, and when the persecution ended the church had to decide what their standing was. In North Africa a rigorist party held that a bishop who had handed over the books had forfeited his office, and that sacraments performed by such a man, including the ordination of other bishops, were void.
The dispute became a schism when the election of a bishop of Carthage was contested on precisely this ground, and the rigorists elected their own, from whose successor Donatus the movement takes its name. Councils, an imperial hearing and eventually force were used against them, and none of it worked. The Donatist church remained the majority in parts of North Africa for a century and was still there when Augustine of Hippo spent much of his career arguing against it.
The theological question the schism forced is one the whole Western church answered through it: whether the efficacy of a sacrament depends on the worthiness of the minister. The answer that prevailed, that it does not, because the sacrament is Christ's act and not the priest's, is now standard in Catholic, Orthodox and most Protestant teaching. It was worked out because a persecution had left a church full of clergy whose conduct under pressure was known to their congregations.