If the signs meet the Ohio ordinance for such (placement, size), they carry the force of law. Not so in Michigan.
And as I've said numerous times, I have never seen a no-firearms sign on a door of a hotel. I've stayed in hotels in Indiana, Kentucky, Tennessee, Ohio, Pennsylvania, West Virginia, Virginia, North Carolina, South Carolina, Florida, Georgia, Alabama, Mississippi, Missouri, Kansas, Texas, New Mexico, Arizona, California, Washington, Wyoming, Nebraska, South Dakota, Wisconsin, Nevada, Utah, Colorado, Maine (might have missed a few) as well as Michigan.
I travel extensively every year both on vacation and for work.