Yes, you need tyre dust caps. Although small, their main purpose is to protect your tyre valves from the dirt of the road. Especially if you are driving through puddles, mud, sand.
They also keep out excess levels of moisture and any other contaminants that could damage your tyre's valve.
To a lesser extent they play a part in preventing air from escaping from the tyre. Protecting the top of the valve from any foreign objects.
Can you drive without dust caps? While it is possible to drive without dust caps, it's not recommended. You are likely to have dirt and moisture build up on the tip of your valve stem, which will damage it.