For your situation (not a resident of an Ohio city*), you do not need to be concerned with city income tax, if you are a W-2 employee. If you work in the city limits of a city with an income tax, your employer is required to withhold the correct amount and remit it to the city. You cannot (normally) get any of that refunded and you do not need to file a city return.
Longer explanation.
Some Ohio Cities have an income tax. Any Ohio city, with an income tax, is allowed to have withholding taken out of your pay while you worked in that city. Your employer is required to do that withholding. You can't get a refund unless it was improperly withheld, i.e. you didn't actually work there for all or part of the time that your employer was withholding. If you live in a city with an income tax, you are subject to that city's tax as well. The city you live in* will usually allow a credit or partial credit for the withheld tax you paid to the work location city.
235 cities and 331 villages in Ohio have an income tax, including Columbus, Toledo, Cincinnati, and Cleveland. Ohio law requires a flat rate that cannot exceed 1%, unless it is approved by the voters. Ohio local income tax rates range from 0.5% to 3% in Parma Heights. Cities only taxed earned income (wages or self-employment), although some also include rental income.
There are 2 state wide agencies that help Ohio Cities collect City Income taxes. One is CCA (Central Collection Agency) and the other, larger one, is RITA (Regional Income Tax Agency). There is a pull down list of cities, in Turbo Tax(TT), for both City Tax Forms (CCA) & RITA. TurboTax can only do RITA & CCA cities. But many Ohio cities use neither agency. These cities design their own forms and require you to use them. You can get paper copies from the individual city. Most are available at the city’s web site. Many cities will prepare the tax return for you, but you have to go to city hall in person. Some may have on line filing (try the city web site). Although they won't usually admit it, some will even accept CCA forms. So you could try filling out a CCA form on TurboTax and mailing it to your city. But, I actually find it easier to fill out a city return by hand or at the city, CCA or RITA web site, than to use TT. Things don't happen automatically and you need to know what you’re doing. The actual city forms are easier to follow than the TT interview.
TT cannot e-file an Ohio City or RITA return. It must be printed and mailed.
*Since you have a permanent residence somewhere else (South Carolina), any housing you stay in while in Ohio does not count as being a resident of that city, for municipal income tax purposes.
Ohio also has School District Income Tax (SDIT). Only residents of a School District (SD) pay that tax. Working in the limits of an SD with a tax does not subject you to SDIT. If your employer mistakenly withholds SDIT (unlikely), you can file an SD tax return for a refund.
