The short answer to a very long and complicated question is, "yes, but only if you owner's manual specifies unleaded premium fuel required". If your owner's manual states that "unleaded premium is recommended", it's your choice; using it is a good idea, but not using it isn't likely to cause engine damage. If your car only calls for unleaded fuel, there is no advantage to using premium unleaded fuel.
If you tow a trailer, drive in extreme heat or live in the mountains, you should always use the fuel recommended by your vehicle's manufacturer. In these conditions, "unleaded premium recommended" really means "unleaded premium only".
You can also find you car's fuel grade requirement inside of the fuel filler door.