- It costs a dollar or more.
- It costs a dollar, or it costs more than a dollar.
Do they have the same meaning?
Yes, they're the same.
"It costs a dollar or more" means exactly one dollar, or an amount more than one dollar. Amount ≥ $1.
"It costs more than a dollar" means amount > $1, but not equal.
Bear in mind that someone may be approximating. If you asked someone the price of a product and he said, "a dollar", I wouldn't be surprised if it was actually 99 cents or $1.05.