We are having very cold weather right now here in Ohio, temps are between 20-30 degrees not counting the wind chill. Should I throw a tarp over my compost pile to help hold the heat in or leave it open to allow air and moisture to enter freely?

Compost is low maintenance in cold weather. It doesn't need to be covered. If you know cold weather is coming you might want to go out with a spade fork and turn it a little, kind of nudge up the activity, and you could put some water on it before the air freezes and the compost will perk right along. If you really want to cover it with something, I would spread straw or hay over the top. It'll break down over time, but will act like a thatch to insulate a bit.

It's always an interesting sight in freezing weather to see the steam rising off of the compost.

I'll take your advise and not cover the pile. Thanks for your response.