"These shoes are not old."
No, じゃない is for -な adjectives. It's also fairly casual. When you want to be polite, you should use ではない (little bit formal), じゃありません (moderately formal), or ではありません (very formal).
古い (ふる.い) = -い adjective = 古くない (not old)
必要 (ひつよう) = -な adjective = 必要じゃない (not necessary)
(Realistically speaking, you could use it じゃない with an -い adjective and people would know what you meant, but it's not "proper" grammar and you'll probably be corrected if people are of the mind.)