If you are male, there is no state law prohibiting you from walking areound with your shirt off. However, there might be specific city ordinances defining when you must wear a shirt (for example in a restaurant).

Laws are more stringent for females. Arizona law (ARS 13-1402) about 'indecent exposure' prohibits females from exposing 'the areola or nipple of her breast or breasts and another person is present.' Indecent exposure is a class 1 misdemeanor and indecent exposure to a person under the age of fifteen years is a class 6 felony.