In "The Minister's Black Veil", the Reverend Mr. Hooper begins to wear a black veil that obscures most of his face, suprising and confusing his congregation. It is strongly suggested that Hooper is wearing the veil because of a woman who has recently died; at her funeral, the following transpires,
The clergyman stepped into the room where the corpse was laid, and bent over the coffin, to take a last farewell of his deceased parishioner. As he stooped, the veil hung straight down from his forehead, so that, if her eyelids had not been closed forever, the dead maiden might have seen his face. Could Mr. Hooper be fearful of her glance, that he so hastily caught back the black veil? A person who watched the interview between the dead and living, scrupled not to affirm, that, at the instant when the clergyman's features were disclosed, the corpse had slightly shuddered, rustling the shroud and muslin cap, though the countenance retained the composure of death.
It seems very clear that the author wishes for us to suspect some connection between the wearing of the veil and the deceased woman.
In Poe's commentary, he speculates that the woman and Mr. Hooper committed adultery. From the standpoint of a dramatic, tragic story, this makes a lot of sense; adultery is, or at least was, in Hawthorne and Poe's era, a shocking crime, and it would make sense for someone with such an immense degree of personal responsibility and morality as a minister to be horribly scandalized by committing it. This is what Poe is referring to as a "crime of dark dye".
Poe chose the phrase "dark dye" carefully; he is implying that the body or soul is like cloth, which can be colored through the use of dye. Sins are often compared to black or discolored spots upon the soul; thus a crime of "dark dye" would imply one that has seeped through and visibly marred the fabric of one's nature, and is difficult or impossible to wash out; the minister will bear this mark for the rest of his life, and indeed he does.