My answer is somewhat complementary to Arnold Neumaier's. The cluster decomposition principle for the S-matrix can be derived from microcausality and the relativistic spectrum condition that defines the vacuum state (Lorentz invariance is actually not necessary), assuming that the vacuum is a *pure* state (which amounts to saying it is the *only* translation invariant vector state in the vacuum Hilbert space). More precisely, in this case one can show by means of the Jost-Lehmann-Dyson representation for truncated correlation functions that truncated vacuum expectation values must have exponential (Yukawa-like) decay in spacelike directions for theories with a mass gap; the same argument leads to a power-law (Coulomb-like) decay in spacelike directions in the absence of a mass gap. When you plug this result into the LSZ reduction formula, you get the corresponding cluster decomposition for the S-matrix.

More importantly, the above cluster property of truncated vacuum expectation values is crucial for the *derivation* of the LSZ reduction formula when the theory has a mass gap (see, for instance, H. Araki, "Mathematical Theory of Quantum Fields", Oxford (2000) or M. Reed and B. Simon, "Methods of Modern Mathematical Physics III - Scattering Theory", Academic Press (1979), section XI.16). If there is no mass gap, stricly speaking the formula no longer holds due to infrared divergences (the Coulomb-like decay is too slow for the formula to hold, as it happens in non-relativistic quantum mechanics with long-range potentials like Coulomb's). In this case one must "fix" the reduction formula by resorting to a "zero-recoil" approximation for soft processes (Bloch-Nordsieck, etc.), which forces us to restrict to the domain of physical validity of this approximation, i.e. infrared-safe processes. A mathematically rigorous justification of this approximation is to date still lacking, though.