Here's the generic answer:
You normally only have to file a return in the city you live in. It is not necessary to file one in the city you work in, if different. Your employer will remit withholding to the city you work in, but you do not file a return unless you're claiming a refund or your employer didn't withhold enough ( both highly unusual). The city you live in will allow a credit or partial credit for the withheld tax you paid to the work location city.
So, yes, if he physically worked in Dayton (not just that the employer has an office there), he is suppose to file a Dayton return to pay the shortfall. But, I wouldn't do it. It's highly unusual for an employer not to withhold the correct amount. There may be some other reason for nothing being withheld. You may want to ask them.
The City of Amherst tax rate is 1.5% with a 1% credit given for taxes paid to other cities. The Dayton tax rate is 2.25%.
