Q. Do I need to claim the apartment?
A. No. As a W-2 employee, you are not allowed(since the 2018 tax law) any business deductions. However, your employer as allowed to reimburse you on a tax free basis. (under an "accountable plan")
Q. Do I also need to file in Illinois even thought the company I work for is paying me directly from Ohio to be on-site in Illinois?
A. Yes, since you physically work in IL. The apartment is irrelevant, you should have been filing an IL non resident return for the last 2-3 years. You also file an Ohio resident return and pay OH tax on all your income, regardless of where earned. OH will give you a credit (or partial credit if IL has a higher tax rate) for the tax you pay to IL.
