No exactly answer for which verb word needs be or which needn't. Here, "be needn't " is also wrong, you could say "be not needed". The question is not which verb word needs "be" or not. The question is you should know that one basic sentence only needs one verb to be predicate, if there're other verb words, they can't be predicates. You should change the verb form as it should be, whatever ,they cannot be predicate form.
