"I have no idea" is correct. Example: When Bill was asked who took the food from table, he replied "I have no idea".
"I have any idea." is not correct. You could, however, say "I don't have any idea" (I do not have any idea). You will also hear, quite often in the southern regional English "I don't have the slightest idea", which also means the same thing. All are just polite ways of saying: I don't know.