What is ironic about fortinbras becoming the new head of government?

Fortinbras can be considered as the antithesis of Hamlet. While the former is careful with the decisions he makes and actions he takes, Fortinbras is impulsive and highly determined to avenge the death of his father by any means possible.

There is irony in the unfoldment of events in the story because Fortinbras had been preparing himself for an eventual engagement in combat in order to avenge his father. However, this never really happened as he finds the throne vacant. On the other hand, Hamlet was initially reluctant of pursuing the throne but ultimately decides to avenge his father's death and claim the throne but dies in combat.