When and why are effective Hamiltonians used? I'm wondering, are there general physical principles behind writing down quantum evolutions in terms of effective Hamiltonians? I'd love a kind of big-picture explaination of their use.
For example, is it like a of Born-Oppenheimer approximation that gives an explicit separation of time-scales between a slow and fast dynamics? Or does it allow the encoding of non-unitary dissipative processes in a Von Neumann time-evolution without resorting to master equations?
 A: Effective Hamiltonians are used when it is practically impossible or impractical to deal with the "true" dynamics describing the system. This may be due to the big number of constituents of the systems, or to the fact that the "true" dynamics takes into account some effects that are too small to be relevant in the scale under study. Typical examples are the mean field effective description of many-particles systems, the non-relativistic description of motion, the classical mechanical description (as an effective approximation of quantum mechanics), ...
I put "true" above within quotation marks because often you already start with an effective description, and make a successive approximation to obtain another one, that is more suitable for the problem at hand.
Of course when you use an effective model, it would be suitable to prove that it comes indeed from a more complex theory by means of some approximation, and also to know how good the approximation is (i.e. to have a bound on the error that you make in doing the approximation). Otherwise, I would say that you have a phenomenological/empirical description rather than an effective description. This distinction is essentially a matter of semantics: until the advent of quantum mechanics, classical mechanics was an empirical theory that agreed with experience concerning the motion of bodies; after the formulation and experimental confirmation of QM, it is rather considered an effective theory that describe well only the motion of "macroscopic" bodies.
The mathematical physics community is often very active in providing justifications for effective descriptions: you will find plenty of works that rigorously prove how mean-field, classical, non-relativistic dynamical theories can be obtained by more fundamental theories by means of suitable approximations, that also provide bounds on the approximation error when possible.
