The main conflict in this story may not be the most obvious one, but it is past versus the present.
Miss Emily is part of the antebellum South. Her home "had once been white, decorated with cupolas and spires and scrolled balconies" but now it was "an eyesore among eyesores."
In the old days, people looked out for one another. When Miss Emily must accept charity, a story is concocted to keep her from being taxed. But, "(w)hen the next gerneration, with its more modern ideas became mayors and aldermen, this arrangement created some little dissatification."
Faulkner carefully let's us see that the past isn't all good, the present not all bad. The slavery issue is of course, wrong, and Miss Emily's family complicit in its continuation. It is a good thing for it to have ended.
The conflict in the present, now, is that people have comes so distanced from one another. Faulkner makes the whole town complicit in their ignoring of human need and kindness. For example, in Part IV: So the next day we all said, She will kill herself"; and we said it would be the best thing."
There are other conflicts in the story: Miss Emily vs. Homer, Miss Emily vs. her own past, loneliness and human need. But the past and the present, for me, is most compelling.
There are three main social conflicts in Faulkner's short-story: cultural, generational, and socio-economic conflict. These conflicts are all present in Emliy's situation regarding her taxes. You can also see these conflict through out the story and don't forget the importance of the symbolism used through out the story; it is very contexual and helps the reader find the theme or moral lesson of "A Rose for Emily."
There isn't a central conflict in this story, because it is an "Absurd Plot." The narrator takes us from the present into the past to explain how the people in this town looked upon Emily, and provides the reader with some back ground on Emily's family. But there are no clear answers given as to why Emily did the things she did or was the way she was. The central conflict would have been, "Why did she kill Homer?" or "Why was Emily so secluded?" The reader can draw conclusions with the information provided, but yet there is no clear reason stated in the story as to why she did this. Her killing Homer only gives insight to why she kept herself secluded, but even before Homer's arrival in the story, Emily was very private and secluded. This is one of those stories that keeps the reader thinking and drawing their own conclusions.
This story would be better used in providing teaching on analyzing setting.