A reference for Calculus of Functors for Model Categories

In Calculus III and its predecessors I studied functors from Top to Top and a few related cases. The ideas clearly generalize to functors $C\to D$ between model categories satisfying some pretty weak axioms, but I did not try to find the right axioms, and I don't think anyone has ever written anything definitive about that.

Is that what you are asking about?

The paper mentioned by Tilson takes the ideas in a somewhat different direction, I think: it's about treating the categories of functors themselves as model categories and finding Quillen adjoint pairs that refine my ideas.

I haven't read it, but maybe this could be useful: http://arxiv.org/abs/math/0601221 Calculus of Functors and Model categories by Biedermann Chorny and Roendigs

You might look at Thomas Goodwillie's papers "Calculus I" (K-Theory, 4), "Calculus II" (K-Theory, 5) and "Calculus III" (Geometry and Topology, 7).