Do Zombies have to pay Inheritance Tax?

No

Inheritance Tax is not payable by the living dead. Apart from life time transfers Inheritance Tax is only payable on death. Death is the complete cessation of all biological functions.
      The ability to walk about groaning and eating human flesh is clearly a biological function.
     If you remove the flesh eating bit, walking and groaning is a common phenomenon amongst men. Often greeted by a total lack of sympathy and much rolling of eyes from women.