"When it was at length conveyed to a Government mint and examined, it was valued at nearly twenty thousand dollars in American money. A great sum for one old priest to have scraped together in a country parish down at the bottom of a ditch" (174) (situational irony)
At Father Lucero's deathbed, he reveals where he has hidden the money that he has been hoarding. It is discovered that he has indeed accumulated quite a substantial amount, though this still seems ridiculous in comparison to the lowly, wretched condition in which he lived his life.
"This gossip did not mean that her servants were disloyal, but rather that they were proud of their mistress" (178) (verbal irony)
The gossip about Doña Isabella's infidelity, rather than impugning her character, actually seems to express a certain incredulity about her youthful beauty. Thus, while taking the language of something that would be socially and morally condemned, such talk actually reveals itself as a kind of exaggeration for the sake of saying the opposite.
Padre Martinez' death (dramatic irony)
Although Padre Martinez is an extremely vigorous, charismatic, and garrulous man, his death is accorded little more than a brief description saying that he died from illness. This is especially ironic since it was in the middle of his schism from the Catholic Church, at which point he seemed to be at the height of his powers. Rather than depicting some showdown between Bishop Latour and this schismatic Mexican priest, Cather simply resolves the conflict, perhaps the most fraught political matter directly told in the novel, by having the latter die.
The allusion to Fenimore Cooper (dramatic irony)
In the prologue, the Spanish cardinal admits that all he knows about America he knows from reading the novels of James Fenimore Cooper, who, in such books as The Last of the Mohicans, romanticized the frontier life and the non-European peoples whose ways of life were being wiped out. In making this literary allusion, Cather is acknowledging her predecessor while also bracketing him as having a limited perspective and a different context that would make his fiction inappropriate for an understanding of New Mexico -- an understanding which she aims to achieve in her own novel.