The fact that Miss Emily Grierson does not pay taxes can be found twice in the story: near the beginning, and then toward the middle.
First, the townsfolk narrator tells us, with a hint of disdain over the fact, that Colonel Sartoris began to remit Emily's taxes as early as 1894, after the death of Miss Emily's father. Keep in mind, that Emily's father and Colonel Sartoris have a past together that could have stemmed from anywhere, from their participation in the war, to their mutual dislike of black people, or perhaps even a faraway tie to the KKK (this is just a speculation but it is a very possible reality of the time). Moreover, Emily's father was a community fixture whom everyone knew for his "mightier than thou" attitude, and for his imposing figure. This is the man who essentially stunts Emily to the point of turning her into a lifelong antisocial woman, which always precludes her from mixing with the "common folk." Either way, Sartoris feels pity for Emily and disguises his intentions to her. He says to her that she does not have to pay taxes after the death of her father.
"from that day in 1894 when Colonel Sartoris... remitted her taxes, the dispensation dating from the death of her father on into perpetuity."
Now, notice that it is a mistake to say "Emily did not have to pay taxes." While this is an approximation of the truth, the reality is that there was a "dispensation" that dated "from the death of her father on into perpetuity." Hence, the only reason she could enjoy this benefit stems from the passing of her father. The dispensation was perhaps a relief for her, since the man only left her the house in which she lived, and nothing more.
Aside from this fact, a mystery continues to lurk beneath the surface. The strange date of the dispensation shows that Emily's father may have owed money to the city. For this reason, once he dies, Emily will not have to pay for the debts that he left behind. Essentially, the government will take whatever they can to make up for the debt. Why make Emily pay, on top of that, when she has no money?
"Only a man of Colonel Sartoris' generation and thought could have invented it, and only a woman could have believed it."
Therefore, upon the death of his father, Sartoris tells Emily that she does not have to pay taxes, and he gave her a false reason, which she took at face value. Again, Sartoris mainly does this out of pity. He knew Grierson well enough to know what the man would have been up to.
Interestingly enough, we learn that after the death of Homer Barron, Emily's taxes are still being remitted, even though Colonel has been long dead since the "dispensation" was "into perpetuity," or perpetual, forever.
"She no longer went out at all... meanwhile her taxes had been remitted."
Current leaders of Jefferson see in Emily exactly what the townsfolk saw: that she is a "duty" of the town, a responsibility that they all need to maintain, together. Yet, they are tiring of the burden she represents as the costs of town government increase, and they want her to start paying taxes, which is the reason for their periodic visitations to her. Emily, however, still symbolizes the notion that she has paid her dues to Jefferson by remaining there as a "fixture from the past." Plus, she was pitied by many because of her social and financial condition.