Here's why:
* "Duke" is a title, not a specific person. There are many dukes around the world, and their customs vary.
* Crowns are typically worn by monarchs (kings and queens). Dukes are usually lower in the hierarchy than monarchs.
To answer your question, you need to tell me:
* Which duke are you talking about?
* What country are you referring to?
For example:
* "Does the Duke of Cambridge wear a crown?" The answer is no, because he's not a king. He's a prince, and he only wears a crown at special occasions, like his coronation if he were to become king.
* "Does the Duke of Burgundy wear a crown?" The answer is no, because the title of Duke of Burgundy has been historically held by members of the royal families of France and Spain, and they are not typically crowned.
Let me know more about the duke you're asking about, and I can give you a better answer!