What does William Faulkner mean by "a sort of hereditary obligation upon the town"?
As this story begins, we learn that Miss Emily did not have to pay taxes to the town in which she lived, and this is the "hereditary obligation" (1) that the town undertook. If a person does not pay taxes, it is up to the rest of the community to make up for this loss of income. This obligation was based upon a declaration by the mayor of the town, Colonel Sartoris, after the death of Miss Emily's father in 1894, "the dispensation dating from the death of her father on into perpetuity" (1).
We are never told precisely why Colonel Sartoris has done this, only that he manufactured a story about Miss Emily's father having loaned money to the town and this being a means of the town repaying the loan. We can infer from the story that Miss Emily's father was a wealthy and influential man since the description of the house tells us that it was a mansion, described as being on "our most select street" (1). It is possible that Colonel Sartoris felt an obligation to Miss Emily because her father had done some things for the town. It is possible that he had served in the war with her father and felt some sort of soldierly camaraderie. Or it is possible that he simply felt sorry for Miss Emily because her father had chased away all possible suitors and she was left with no inheritance but the house. It is not likely that she could have afforded to pay any taxes at all.
What we do know is that the subsequent generations, following that of Colonel Sartoris, felt no such obligation to Miss Emily, and the town began to demand its taxes. He had never documented this dispensation, and she began to receive tax bills. A committee of townspeople was sent to the house to meet with her and explain her tax debt, but Miss Emily denied this obligation and sent them away, managing to pay no taxes to the date of her death.