Do I have to file taxes in both CO and IL if I collected unemployment in CO but lived in IL?

Note that the Colorado Form 104PN Part-Year Resident/Nonresident Tax Calculation Schedule 2020 includes 

"Enter income from line 8 that is from State of Colorado unemployment benefits; and/or is
from another state’s benefits that were received while you were a Colorado resident. "

Since you did live in Colorado and had Colorado source income you will need to file a tax return in Colorado.

You must file a Colorado income tax return if during the year you were:

• A full-year resident of Colorado, or
• A part-year resident of Colorado with taxable income during that part of the year you were a resident, or
• A nonresident of Colorado with Colorado source income,

AND

• You are required to file a federal income tax return, or
• You have a Colorado income tax liability for the year.

A Colorado resident is a person who has made a home in Colorado, or a person whose intention is to be a Colorado resident. The department will consider, among other things, Colorado voter registration, Colorado vehicle registration, Colorado driver license, school registration, property ownership and residence of spouse or children in determining intention to be a Colorado resident.