State tax filing

I live in a township no income tax, my job is located in the same township. I’m a garbage man my route that I pick

up for is in Fairfield. I was just told that I have to pay income tax to a city I don’t work for, or live in. Upon looking into their tax laws it says no one pays income tax who don’t work for the city, live in the city or have a business in the city. According to the companies payroll there’s nothing they can do. I’ve been with this company for 5 years and this is the first time it’s been mentioned. The other issue is this, I get paid for 8 hours if done in under 8. When I work ot on the weekend that city gets that money, when I go on vacation that city gets that money. We have many operations in this company from drivers picking up construction boxes and driving back in to dump material that can’t be charged a tax per day since their all over the place and not in one local city. How does that work if 100% of your income has to be taxed when I work the weekend it’s in the township. That should be non taxable income since the township doesn’t have an income tax.