Both Quantum Field Theory and fluid dynamics rest upon discarding finer details of the system and/or small-scale degrees of freedom. I understand that both frameworks require such removal phenomenologically, but what are the requirements for the Hamiltonian/Lagrangian densities in order for this to be possible?
I guess that either (or both) locality of the interaction and superposition principle play some role here, but I know no method of proving it.
This is a very general question, and I am not sure whether there is a simple way to answer it.
1) Fluid dynamics: The need to have scale dependent parameters can be seen purely within fluid dynamics, without making explicit reference to an underlying theory. Fluctuation-dissipation relations require that dissipative fluids have thermal fluctuations (just ordinary thermodynamics requires that, too). I can integrate out thermal fluctuations at short scales, and I discover that this corresponds to a renormalization of the tree level parameters. The only loop-hole is that, possibly, fluid dynamics was not the correct long-distance description in the first place. This is certainly possible, for example if the underlying theory does not thermalize (is not ergodic). Examples are integrable classical or quantum systems. It also happens in practice if the underlying theory only equibilibrates on absurdly long time scales (as in granular fluids, for example).
2) (Effective) quantum field theories: Again, I can study scale dependence purely within my EFT, without reference to the precise underlying theory. Just study the behavior of the effective action under mode elimination. Again, the main loop-hole is that maybe my theory is not described by an effective quantum field theory. I think this is unlikely; books like Weinberg's QFT book are devoted to showing that basic principles like unitarity, locality and cluster decomposition inevitably lead to QFTs.
