Question: Homological algebra for abelian groups is a standard tool in many fields of mathematics. How much carries over to the setting of commutative monoids (with unit)? It seems like there is a notion of short exact sequence. Can we use this to define ext groups which classify extensions? What works and what doesn't work and why?

Answer: 
Your question can be understood as how to do Homological Algebra over the Field with one Element.

Deitmar, in <http://arxiv.org/abs/math/0608179> , section 6, gives an example of what can go wrong if you try to do sheaf cohomology directly via resolutions...

You might also want to look at his <http://arxiv.org/abs/math/0605429> ; in order to construct K-theory of monoids he sets up an analogue of the Q-construction. The Hom-sets in the resulting category are sort of Exts, maybe something to start with...

Durov, in <http://arxiv.org/abs/0704.2030> , follows the simplicial approach for commutative monads, of which commutative monoids are a special case
I can't give a substantive answer to this question, but differential graded abelian monoids have been sighted in the wild here:

A Chain Functor For Bordism 
Author(s): Stanley O. Kochman 
Source: Transactions of the American Mathematical Society, Vol. 239 (May, 1978), pp. 167-196 
Published by: American Mathematical Society 
Stable URL: <http://www.jstor.org/stable/1997852>