No, you must apply in the country where you are normally resident.
You should only apply for a passport at Consulate of Ireland in Chicago if you reside in Alabama,Arkansas, Illinois, Indiana, Iowa, Kansas, Kentucky, Louisiana, Michigan, Minnesota, Mississippi, Missouri, Nebraska, North Dakota, Ohio, Oklahoma, South Dakota, Tennessee, Texas, or Wisconsin, or in the case of a lost/stolen passport while visiting Chicago, where a replacement document is required for onward travel.
If you live in states other than those listed above, you should apply to the Consulate of Ireland that covers your state. Residents in western states should apply to the Consulate of Ireland in San Francisco, while residents in eastern states should apply to the Consulate of Ireland in New York or Boston. See here for further information.