In Shakespeare's King Lear, the Duke of Burgundy faces a significant dowry-related issue concerning his courtship of Cordelia, Lear's youngest daughter. Initially, the Duke is interested in marrying Cordelia, but he is put in a difficult position when Lear decides to disinherit her after she refuses to flatter him with false declarations of love, unlike her sisters Goneril and Regan.

The dowry issue arises when the Duke of Burgundy, who is vying for Cordelia's hand, is concerned about the implications of Lear's decision. He is interested in marrying Cordelia not only for her beauty and virtue but also for the wealth and power that come with being Lear's daughter. However, when Lear disowns Cordelia, the Duke's interest wanes because the political and financial advantages of marrying her diminish significantly.

Ultimately, the Duke of Burgundy's proposal is overshadowed by the events that unfold, particularly when Cordelia is left without a dowry and is subsequently courted by the King of France, who values her for her character rather than her wealth. This shift highlights the themes of loyalty, love, and the transactional nature of marriage in the context of power and wealth in the play.

In summary, the Duke of Burgundy's dowry-related issue is tied to the political and financial implications of marrying Cordelia, which are complicated by Lear's decision to disinherit her.