In the Ming Dynasty, could the hereditary heirs of titles such as Duke and Marquis (that is, the crown prince) take part in the imperial examination?

No, I clearly tell you that you can’t. Because of the Jingnan Campaign, the Ming Dynasty prohibited relatives of the emperor from participating in the imperial examination system in order to prevent the expansion of aristocratic power.

In December of the 23rd year of Wanli (1595), it was stipulated that the descendants of the clan were allowed to participate in the imperial examination system. As a Confucian scholar, he could participate in the imperial examination, and those who passed the examination could obtain birth qualifications, but this was not implemented in the Wanli Dynasty. It was not until the fourth year of Chongzhen (1631) that the children of the clan actually took part in the imperial examination, and some of them obtained the qualification of Jinshi.

That is to say, these people were not allowed to participate in the imperial examination until the end of the Ming Dynasty. In the Ming Dynasty, most of the titles such as Dukes and Marquises were hereditary from the founding generals. In addition, most of them were married to the royal family, so I speculate that it is unlikely that they would participate in the imperial examination. In addition, most of them had preliminary qualifications. You can directly enter the officialdom or the military system, and there is no need to take the imperial examination.