Without AC, it is impossible to prove that every set is equipotent to an ordinal (in ZF !) and it is impossible to prove that a functor is an equivalence of categories if and only if it is full faithful and essentially surjective ; so I guess that  a lot of things are wrong without AC, in particular concerning combinatorial model categories because very often we have to use transfinite cardinals. For example, "[Implications of large-cardinal principles in homotopical localization][1]" or "[Definable orthogonality classes in accessible categories are small][2]" for links between large cardinal axioms and Bousfield localization.


  [1]: http://www.sciencedirect.com/science/article/pii/S000187080400297X
  [2]: http://arxiv.org/abs/1101.2792