Test each option for its truth value directly.
(A) claims zero is prime. A prime number has exactly two distinct positive divisors, and zero has infinitely many divisors, so (A) is false.
(B) claims one is an integer, which is correct, so (B) is a true statement.
(C) claims 31 is a whole number but not prime. Since $31$ has no divisors other than $1$ and $31$, it is prime, so the second half of (C) is wrong, making the whole statement false.
(D) claims $-31$ is an integer but not a whole number. Whole numbers start from $0$ and go upward, so a negative number cannot be whole, so (D) is fully correct.
Only (A) and (C) turn out false, so those are the ones that are not true.
\[\boxed{\text{(A) and (C) only}}\]