Emily Grierson dies of an unnamed illness in "A Rose for Emily." Without a doubt, her advanced age also contributed to her end. She lived to be quite old, to the point where the town she lived in considered her a living relic of the Old South (she is even referred to as a "fallen monument" by the narrator).
Emily slowly loses contact with the townspeople over time. For a while, people send their children to her for china-painting lessons, but this business eventually wanes. It gets to the point where the people only see Emily through the windows of her crumbling old house. By the time her death is announced publicly, everyone is shocked, since they were not even aware she was ill.
The conditions of the house might have also contributed to Emily's death. The interior is described as dusty. Even the pillow on which she passes away is "yellow and moldy from age and lack of sunlight." Such conditions are symbolic of Emily's own stubbornness regarding change: she refuses to pay taxes and refuses to have numbers put on her mailbox. Even her love is possessive and unchanging, as is seen in her reluctance to give up her father's corpse and her keeping the corpse of Homer Baron well after she kills him.
Overall, Emily's illness and her advanced age are what led to her death, but living in a rotting house may have also contributed to this.