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.