|“||Why so serious?!||„|
|~ The Joker|
It is almost impossible for a Big Bad to be a Comic Relief, because if the main antagonist is comical, then by default most of the story's other elements will be comical as well and this trope cannot apply to the work. The only exception would be if a story primarily driven by realistic drama inexplicably had a comical or cartoonish bad guy who was the only recurrent antagonist but who was irrelevant to the rest of the plot.
Comic reliefs can have their more threatening moments (such as when their true intentions are revealed, i.e. GLaDOS) but their comedy must be more prominent and/or remembered.
Pure Evil villains can never qualify because their acts are always serious and fearsome. This is applicable even if they appear to be funny because their humor is usually sadistic and makes them even more threatening, while a Comic Relief's humor is genuinely played off to make them less threatening.
Comedy villains can never qualify unless there is a game changer in the story the villain is in, considering if a villain is in comic relief, it means the story is very serious but this villain lightens up the story, while comedy villains are in a story that is very comedy intended.