The negation of the axiom of choice only allows us to prove that there is some set which cannot be well-ordered. There is some family of non-empty sets whose product is empty. There is some partially ordered set in which every chain is bounded, but there is no maximal element. And so on.
Of course, from a family without a choice function we can easily construct a set which cannot be well-ordered, and with it a partial order witnessing the failure of Zorn's lemma, and other examples. But we cannot really say much about this family. Is it a family of finite sets? Is it a family of countable sets? Is this family well-orderable? And so on.
It turns out that mathematics is not just "you either love someone or you hate them". If you have no choice, you can still have plenty of degrees of choice, and without pointing out how much choice you have, or don't have, it's very hard to say much.
Furthermore it is possible that the axiom of choice fails so very very far up the cumulative hierarchy that no set used by any mathematician (except set theorists, maybe) is a witness for this failure. In such universe it is true that some theorems will fail (e.g. there will be a commutative unital ring without a maximal ideal, and there will be a vector space without a basis), but their failure occurs so far beyond our interest that it's just as well possible to assume that it doesn't happen.
All we can say, in case we assume the negation of AC, that all those principles equivalent to the axiom of choice fail somewhere, but we cannot possible give an intelligent answer about where these failures occur.
You may also be interested in this math.SE answer of mine, which discusses a similar question.