Tragedy
Obviously, as a play that was written for theatrical performance, Death and the King's Horseman definitely fits into the drama category.
However, insofar as this drama is filled with drama, there are tragic elements at work here as well. We have a hero who is initially revered and destined to achieve everlasting glory by making a monster of a sacrifice, and then totally fails—partially through his own weakness (his hesitation can be understood as a character flaw), and partially through the intervention of Oyo's British residents (the kind of overwhelming force that often takes down the hero in tragedies). So this drama's also a tragedy.