There are three kinds of irony found in literature, the first of which is essentially sarcasm, or verbal irony, when someone says the opposite of what they mean. The second is dramatic irony, which usually occurs when the audience is aware of something important to the plot, but a character is not. The third type is situational irony, when what is happening is the opposite of what one might expect.
In this poem, we can find situational irony in the discrepancy between how society wants the speaker to act and how she actually acts. While society might tell "twisted lies" about the speaker—a black woman —the speaker, far from being downtrodden, is determined to "rise" instead. Her metaphorical upward motion is the opposite of the expected capitulation. The speaker uses "sassiness" to "upset" those who would see her "broken" and behaves instead as if she has her own "oil wells," "diamonds" and "gold mines"—symbolic of wealth.
The speaker's behavior, then, is that of a positive, upbeat person who is sure of her own worth, which is ironic because those around her would rather "shoot" and "cut" her down to size.