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: 
I used to think about this problem in relation to a chain theory for **bordism** ( as mentioned by Josh Shadlen above).  

The problems you have with monoids is first and foremost that the category is not balanced. That means that you can have an epimorphism that is also a monomorphism but NOT an isomorphism. eg. the inclusion of N --> Z.   

Subsequently most constructions that you would like to make - notably short exact sequences and the snake lemma - fail at some level.  

I made a few notes on this as part of my investigation into bordism theory / homework assignment
 [here](http://www.sirarnold.net/maths/homework_bordism_and_homological_algebra_for_monoids.pdf).

In this case, we have complexes of free abelian monoids whose homology takes it's values in abelian groups and yet, the long exact sequence does not come from a short exact sequence of monoid complexes.  

My references include:  

[Bau89] Friedrich W. Bauer. Generalised homology theories and chain complexes. Annali di Mathematica pura ed applicata, CLV:143–191, 1989.  

[Bau95] Friedrich W. Bauer. Bordism theories and chain complexes. Journal of Pura and Applied Algebra, 102:251–272, 1995.  

[BCF63] R.O. Burdick, P.E. Conner, and E.E. Floyd. Chain theories and their derived homology. Proceedings of the AMS, 19(5):1115–1118, Oct. 1963.  

[Koc78] S. O. Kochman. A chain functor for bordism. Transactions of the American Mathematical Society, 239:167–196, 1978.
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>