Special licenses for HVAC contractors (heating, ventilating, and air conditioning) are not required by the state of Illinois, but local government may have their own requirements. These requirements are sometimes difficult to find online, and your best bet is to contact your local government directly.
Go to the online list of Illinois county websites (see Resources for link).
Follow the link to the website of the county in which you wish to conduct business.
The county websites vary, but all of them will have contact information either in a "Contact Us" or similar section, or listed at the bottom of the page. Contact your local administration and inquire about HVAC license requirements.
You may also use a site like BusinessLicenses.com to quickly find and print local business license requirements, but this will cost money and is not as reliable as directly contacting your government. Nevertheless, the link to the site's Illinois counties list can be found in the Resources section.