One final note: I think Faulkner might argue that traditional southern notions of ladyhood necessarily make a woman dysfunctional. Racism, gender roles, and class all interact to construct this southern white womanhood in the form of the lady, and since each ideology is by definition repressive in its restrictions to identity, the final product--the white southern lady--cannot help but be dysfunctional as an identity southern women would need to act out or resist. Hence the problems Emily has with her family, the town, her lover, and her inability to separate from him. She both succumbs to and resists ideologies of her culture, which is why she is so difficult to understand, and why so many readers at once admire her (good for her!) and are horrified by her (she murdered a man and slept with his corpse?).
In addition to Rene's answer, I would say that Emily probably suffers from some parental anxiety issues. Her father, the famous Confederate war hero, has left his daughter with a legacy to which few can live up to, for which his offspring are resented and for which Emily arguably does not deserve.
Further contributing to her "dysfunction" is the alienation she experiences as a refult of her father's legacy. Emily, from birth, is identified as "the other" as "not us." This distinction makes it nearly impossible for Emily to develop real friendships; it is an "us" and "them" mentality over which poor Emily really has no control.
The one way she dysfunctionally tries to gain control is through her captivation (quite literally) of Homer. Rat poison and sleeping with your lover's corpse is not generally considered the picture of mental health...but nothing in Emily's existence has allowed her to experience life as her townspeople do...
Miss Emily has trouble letting people go after they die. She refuses to release her father's body until he has been dead three days. Not to mention her one-time suitor's discovery in her bed-a very decomposed corpse in her bed. It appears Miss Emily has a separation anxiety.