I am not sure that it is ironic to call Chillingworth a leech -- it seems to me more that it's appropriate. But here's the idea:
In those days, doctors were often called leeches because they would put leeches on people to have the leeches suck blood from the people. This was in the days when they believed "bleeding" people could heal them. So Chillingworth is called a leech because of this.
But really, he's bleeding the life out of Dimmesdale. So he's being a leech in a much more malicious way. He is taking the life out of Dimmesdale by hounding him about his dark secret.
Roger Chillingworth, the cuckolded husband of Hester Prynne in Nathaniel Hawthorne's The Scarlet Letter is the greatest sinner of all the characters. For, he enters the home of Dimmesdale under the pretense of being a healer of the body when, in truth, he desires to "violate the sanctity of the human heart" by probing into the conscience of Arthur Dimmesdale. While pretending to be the friend and confidant of Dimmesdale, he seeks to discover if the minister is the man who has sinned with his wife. Like a leech, he attaches himself to Dimmesdale, and through his insidious and surreptitious questions he seeks the knowledge that is in the heart of the unsuspecting minister. His drawing forth from the ailing psyche of Dimmesdale is anything but healing; instead the minister is weakened the longer that the physician dwells with him, growing paler by the day. In Chapter XIV when Chillingworth and Hester talk, he tells her with a fierce pride of his torture of the minister. Describing what has taken place he says,
A mortal man, with once a human heart, has become a fiend for his especial torment.
Chillingworth suggests further that, rather than having paid his debt, Dimmesdale has increased it, by making of Chillingworth the fiend which he has become. He is guilty of what Hawthorne calls "the unpardonable sin": the subordination of the heart to the intellect. It occurs when one is willing to sacrifice his fellow man in order to gratify his own selfish interest. This interest is to destroy Dimmesdale; the promise he made to Hester in their interview of Chapter IV: "He will be mine!"
And, here lies the irony: leeches were placed on ailing people in order to draw out the bad blood which made them sick. Thus, leech became synonymous with physicians of the 17th century since this method was employed often by them. However, this physician, the leech, Chillingworth, drains the very soul of Arthur Dimmesdale in order to learn its secrets, further debilitating him. Like the leech, Chillingworth's soul is the blackest, for his revenge is dependent upon the destruction of another human being, not the healing.
Roger Chillingsworth does actually care for the Reverend Dimmsdale in the book The Scarlet Letter. As the story progresses Roger has had more of an opportunity to see that Dimmsdale is not all that he appears to be. Roger searches and speaks with Dimmsdale. He is curious about what sin the Reverend had committed but unable to find out.
I think that when one looks at the term the leech and tries to find the irony in the title one has to also look at the things that are said about Roger. As much as there is stated about his intent to find out about Dimmsdale's hidden sin, Roger is also presented as a good man.
There the physician sat: his kind, watchful, sympathizing, but never intrusive friend. (113)
Even after he has probed the physician he responds by putting their friendship back together. There is a need for him to find out the secret which makes him grasp onto the Reverend's words, but at the same time there is a friendship that Roger respects. This may be the irony in the term the leech.
However, as the Reverend sleeps, the physician opens his vestments that cover the chest. (The reader does not know at this point that Dimmsdale has a scarlet letter "A" embedded into his own skin). Roger sees it and then he jumps and reaches his hands high above. He is identified next as the devil cheering with joy because he taken a soul.
Roger is a leech that has taken the secret from Dimmsdale.