"They pay me as if I am in Ohio"
That's wrong. You work in Illinois, not in Ohio. It doesn't matter that your employer's headquarters or payroll office is in Ohio. They should be withholding Illinois tax, and box 15 on your W-2 should say IL. If you file your tax returns properly, you will pay most of your tax to Illinois and very little, if any, to Ohio.
If your employer will not withhold Illinois tax, you should be making quarterly estimated tax payments to Illinois.
"additionally I also get lodging and food money back"
How do you get the lodging and food money? Is it a fixed amount, or do you report your actual expenses to your employer? Do you get a separate check for expenses or is it included in your pay?
If the reimbursement is paid through an accountable plan you don't put it on your tax return, and you don't pay tax on it. The plan would have to be established by your employer. An accountable plan would generally require you to submit expense reports with receipts, but that might not be required if the plan meets the requirements for per diem reimbursement. Reimbursement from an accountable plan would not be included in the amount on your W-2.
If the reimbursement is not through an accountable plan, it's just additional pay, no matter what the employer calls it. It should be included in the wages on your W-2 and you pay tax on it the same as any other pay.
