What is an example of a major ethical dilemma between two people in To Kill a Mockingbird?

An ethical dilemma occurs when what's morally right isn't necessarily clear. A person in an ethical dilemma might find him or herself in the midst of a potentially untenable situation, and any decision requires much soul-searching until the most acceptable outcome is reached.  

To Kill a Mockingbirdis full of situations where right and wrong are certainly apparent, and people still make the worst decision as they fail to operate according to a fair and...

An ethical dilemma occurs when what's morally right isn't necessarily clear. A person in an ethical dilemma might find him or herself in the midst of a potentially untenable situation, and any decision requires much soul-searching until the most acceptable outcome is reached.  

To Kill a Mockingbird is full of situations where right and wrong are certainly apparent, and people still make the worst decision as they fail to operate according to a fair and just approach that benefits society.

Dolphus Raymond recognizes the townspeople's flawed moral compass when he says, "Cry about the simple hell people give other people—without even thinking. Cry about the hell white people give to colored folks, without even stopping to think that they're people too..." and he likens Tom Robinson's guilty verdict to the killing of a mockingbird. 

One specific and significant, or major, ethical dilemma in Harper Lee's novel comes at the end when Atticus assumes that Jem killed Bob Ewell. He thinks his son must therefore face the consequences. Atticus is torn apart by his own need to uphold justice, even if it involves his own son. Atticus blames himself for putting his son in that situation and plans to defend him in court. Sheriff Tate is the man faced with the huge burden of removing guilt from Jem, without implicating Boo Radley, himself an innocent victim. Tate knows how Atticus will not accept lies, and he is careful to describe the situation in terms that may omit details but are not blatant lies. (Jem is too small to take on Ewell. Ewell was drunk and carrying a knife, and it was too dark for anyone to discount his version.)

As a man of the law himself, Tate must retain his own honor and must uphold the law. He makes his decision based on his understanding of the situation, his enormous respect for Atticus, and his realization that the townsfolk of Maycomb cannot be trusted to reach the right conclusion. Atticus makes the decision to accept Tate's version of events. Accordingly, Boo Radley receives the protection he deserves.