After her father dies, Emily wants to have total control over her environment. She asserts this control by insisting on stasis: she wants nothing around her to change once she has her life organized the way she wants it.
Emily won't pay taxes after her father's death because her father was promised he would never have to. The town felt protective towards its aristocrats, because these families represented to it their collective nostalgia for a lost way of life. Therefore, the mayor concocted a story that the town owed the Griersons money, and for that reason they were exempt from property taxes.
Emily can no more pay the taxes than her father could—the town's people, in fact, knows she is poor, and look forward to her having to scrimp and save as they do. She is, however, arrogant and rigidly locked into place in her attitudes and ideas. Because her father didn't pay, she sees no reason why she should have to, so she doesn't. The town capitulates to her will.
Her refusal to accept free mail delivery is another sign of her rigidity, arrogance, and refusal to change:
When the town got free postal delivery, Miss Emily alone refused to let them fasten the metal numbers above her door and attach a mailbox to it. She would not listen to them.
Emily kills Homer because she wants to keep him with him her. She realizes she is about to lose him, so she poisons him. This keeps her life the same and keeps her in control of him, even if the "him" she has control over has now become a corpse.
Emily's extreme stasis is a sign of mental illness and symbolizes the "illness" of the South in being unable to move beyond its past.