No, he will only need to file the Illinois state tax return in this case. But, he may have needed to file in New York if his income exceeded the 2020 New York standard deduction.
See New York Nonresident FAQs:
If you are a nonresident whose primary office is in New York State, your days telecommuting during the pandemic are considered days worked in the state unless your employer has established a bona fide employer office at your telecommuting location.
There are a number of factors that determine whether your employer has established a bona fide employer office at your telecommuting location. In general, unless your employer specifically acted to establish a bona fide employer office at your telecommuting location, you will continue to owe New York State income tax on income earned while telecommuting.
For future reference,
Form IT-2104-E Certificate of Exemption from Withholding would be used by college students in limited circumstances.
If he filed, Form IT-2104 Employee's Withholding Allowance Certificate, this would not necessarily eliminate any tax liability. See the instructions under Who should file this form which state in part: ..if the employer does not have Form IT-2104, they may use zero as your number of allowances. This may result in the wrong amount of tax withheld for New York State, New York City, and Yonkers.
