There are a few factors to consider when deciding whether you need special shoes for road biking. The first is the type of pedal you have. If you have pedals with cleats, then you will need shoes with cleats that match those pedals in order to clip in and out properly. Some pedals have built-in cleats while others require separate cleats that attach to the bottom of your shoes; either way, you'll need specific shoes. The second factor is the amount of support and padding you want in your shoes. Road biking tends to be a more aggressive form of riding than, say, cruiser biking or even mountain biking, so having well-padded and supported shoes can help prevent pain and injury. Finally, consider what kind of climate you'll be riding in. If you live in an area with cold winters, then getting shoe covers or investing in a good pair of winter cycling boots can make a big difference in your comfort level while riding.
In short, there's no one-size-fits-all answer to whether or not you need special shoes for road biking. It depends on your individual needs and preferences. However, if you plan on doing any serious road biking, it's probably worth investing in a good pair of shoes specifically designed for the activity.