Can I tarp my car and leave it outside or must I garage it? (Ohio)?

Given the choice of a tarp or a garage, take the garage. A tarp offers less protection and can damage the paint.

A tarp (outdoors) might be OK for a work-in-progress. They will keep the sun off, but collect condensation and can scratch a nice, fine paint job, as the tarp moves with the wind and rubs fine dirt along the car's surface.
Use a garage.

You should definitely store your car in a garage. A tarp will protect it from things like leaves and bird poop, but not much else. The car will still be exposed to the weather (cold, rain, ice, etc), and like the other answers said, the tarp could damage the paint job.

IDK. In AZ. you must buy a regular car cover, a tarp is illegal.

Garages are legal here, too.

is there much benefit to having a garage ???? the muscle between your ears needs a workout

Tarps can do A LOT of damage. You need to carefully look into this topic. The wind moving the tarp will wear out certain area and dirt will get under there and acts like sandpaper.
And as others have said a tarp holds moisture.
A car port or car tent might be an option. Get some local classic car club folks and see what they do. Things vary by location based on weather.

Best to garage it. Also, put a float charger on the battery or it will be dead junk in the spring.

Inside is always better than under a cover as you will be less likely to have condensation form between the tarp and the vehicle. Moisture which will never dry out. In some ways it better to leave the car uncovered. Use the garage is better by far - you can even plug in the battery to a smart charger and not have to worry about it going flat etc. but it depends how much you want your car close to you. If you can't live without It just keep it at your place and take it for a run every couple of weeks.

Can you tarp it? Yes. But anyone with the option of free garage space who even has to ask is a complete dumbass in my book.