Generally, yes for plumbing & electrical work you need to get a permit from Franklin County.

For Plumbing,

Contact the Franklin County Public Health Department

See: https://myfcph.org/environmental-health/plumbing/

Or Call (614) 525-3160

Located at;

Franklin County Public Health
280 East Broad Street
Columbus, Ohio 43215-4562

For Electrical,

Contact the Franklin County Building Zoning Department

See: https://development.franklincountyohio.gov/building-permits

Or Call (614) 525-3095

Located at;

PERMIT COUNTER:
142 S. Front St.
Columbus, Ohio 43215