Verbal Irony:

Can I have these old quilts?

This is ironic for a couple of reasons. First, Dee has asked for quilts that had been quilted by Big Dee and her mother. She has just finished telling her mother that she no longer wants to be called Dee because she needs to distance herself from "people who oppress" her. And here she is, trying to link herself in a tangible way to something Big Dee, whom she was named after, has created. The word "old" is ironic in itself because Dee seems to be making great efforts to distance herself from the "old" in her life: her family, her name, her origins. Her mother has even offered her these quilts before, and she has turned them down.

You just don't understand . . . Your heritage.

This is also an ironic statement because Dee clearly doesn't value her entire heritage. She has been quite pretentious about the parts of her heritage she's willing to recognize, but she thinks that her mother is the one who is blind—simply because she refuses to cave to Dee's wishes.

Dramatic Irony:

The reader is aware that Mama is at least a bit protective of Maggie, who has been burned badly in a fire and doesn't have the confidence of her other daughter. Mama sees the potential in Maggie, while Dee only sees her as having a "brain like an elephant's" (not used in a positive connotation) and calls her "backward." When she asks for Maggie's quilts, she undoubtedly expects her mother to give them to her, but she doesn't realize how Mama feels about her other daughter.

Situational Irony (or Irony of Circumstance):

Dee expects to get what she wants. She surveys her mother's house, laying claim to all the items she's planning to take with her when she leaves. And she is flabbergasted when Mama gives something of value to her sister instead of her: "Maggie can't appreciate these quilts! She'd probably be backward enough to put them to every day use." This shows the contrast in the two daughters, as well. Dee is all about image, while Maggie reflects practicality. The irony is that her mother prefers the practical and remains true to her promises in giving the quilts to Maggie. Dee does not see this coming.

Dramatic irony is present in the story.  Dee was glad her first house burnt down to the ground and as her mother said in reference to the new house, "No doubt when Dee sees it she will want to tear it down." Dee has never wanted to be part of her cultural heritage and left to get away from all the memories of her home, poverty, family, etc.  All of a sudden she returns to grab things that are part of her family only so she can display them to her Black Muslim family that are only interested in what they stand for and not for whom they stand for.  Then as quickly as she pays a visit to the past, she picks up and walks out on "IT" again.  It is obvious, to her heritage is for show not for living.

Source:  The Language of Literature Book by McDougal Littell