Colonel Sartoris has two roles in "A Rose for Emily," one of which is to help indicate Miss Emily's state of mind, and the other is to establish the conflict between the Old South and the New South.
When the aldermen of Jefferson visit Miss Emily in order to convince her to pay her taxes, she responds:
I have no taxes in Jefferson. Colonel Sartoris explained it to me. Perhaps one of you can gain access to the city records and satisfy yourselves.
So far, there doesn't seem to any problem with Miss Emily's conception of time or her state of mind. But, after the aldermen continue to insist that she owes taxes, she dismisses them and tells them to "see Colonel Sartoris. I have no taxes in Jefferson." At this point, we understand that Miss Emily really does not understand or remember that Colonel Sartoris has been dead for ten years--time, which moves on outside of Miss Emily's home, does not seem to move for her. This is perhaps the first sign that Miss Emily holds tenaciously onto the past.
If you recall the story about the remitting of taxes, you'll see that Colonel Sartoris, knowing that Miss Emily is no longer wealthy, makes up the story about her father loaning money to the town and the town repaying the debt by permanently remitting her taxes:
Only a man of Colonel Sartoris' generation and thought could have invented it [the story], and only a woman could have believed it.
Colonel Sartoris (and Miss Emily) are part of the Old South, the south of an aristocratic, land-owning class, which no longer existed after the Civil War, and Faulkner uses Sartoris in this instance simply as an example of what the Old South, as opposed to the unsentimental, democratic New South, represented--respect for, and deference to, the aristocratic stratum of the society and kindness. In addition, Colonel Sartoris, understanding that giving outright charity to someone of Miss Emily's class is inappropriate, provides charity to Miss Emily by creating the fiction that the remitting of taxes is repayment for a debt. She would not have accepted help any other way.
Colonel Sartoris, then, allows Faulkner to establish the fact that Miss Emily has no conception of time moving on and to begin examining the conflict between the Old South and the New South.