Is it correct to say "Whose turn to" instead of "Whose turn is it to"?
"Whose turn to put the rubbish out?"
"Whose turn is it to put the rubbish out?"
The second is clear and grammatically correct. The first is well understood and you will find a discussion of this sort of construct in What does "What price [noun]?" mean? , where parallel questions such as "What price ...?" are discussed.