Elementary particles must satisfy the principles of relativistic quantum field theory. This implies that they are described by nontrivial irreducible unitary representations of the Poincare group, compatible with a vacuum state and causality.
Having a unitary representation of the Poincare group characterizes relativistic invariance. Irreducibility corresponds to the elementarity of the particle. The vacuum is excluded by forbidding the trivial representation.
Finally, causality requires the principle of locality, namely that commutators (or in case of fermions anticommutators) of the creation and annihilation fields at points with spacelike relative position must commute. Otherwise, the dynamics of distant points would be influenced in a superluminal way.
This rules out many of the irreducible unitary representations (completely classified by Wigner in 1939), leaving only those with nonnegative mass and finite spin. Of the other irreducible unitary representations, all of which were classified by Wigner in 1939, the massless continuous spin (also referred to as infinite spin) representations are those most difficult to dismiss of.
On page 71 of his QFT book, Weinberg simply says that massless particles are not observed to have a continuous degree of freedom. Weinberg uses an empirical fact (''are not observed to have'') to eliminate this case in his analysis. He says that there are such representation, but that they are irrelevant as they don't match observation. One can eliminate the continuous spin representation also by causality arguments; but these arguments are lengthy:
Massless particles with continuous spin indices,
Phys. Rev. D 13 (1976), 2291-2294.
Quantization of massless fields with continuous spin,
Prog. Theor. Phys. 58 (1977) 652-666.
But Weinberg doesn't want to do more representation theory than necessary. Since these representations do not lead to causal quantum fields, he refers to experience to be able to take a shortcut.
However, the literature also discusses almost acceptable variations of traditional quantum fields involving continuous spin representations:
Zero-mass infinite spin representations of the Poincare group and quantum field theory,
Annals of Physics 64 (1970), 195-203.
G.J. Iverson and G. Mack,
Quantum fields and interactions of massless particles: The continuous spin case,
Comm. Math. Phys 18 (1971), 211-253.
J. Mund, B. Schroer and J. Yngvason,
String-Localized Quantum Fields and Modular Localization,
Comm. Math. Phys 268 (2006), 621-672.
X. Bekaert and J. Mourad,
The continuous spin limit of higher spin field equations,
J. High Energy Phys. 2006 (2006), 115.
R. Longo, V. Morinelli and K.-H. Rehren,
Where infinite spin particles are localizable,
Note that some higher derivative string theories give rise to particles belonging to the continuous spin representation:
Tensionless strings: physical fock space and higher spin fields,
Int. J. Mod. Phys. A 19 (2004) 3171-3194.
Continuous spin particles from a string theory,
Note also that irreducibility (while characterizing elementary particles) is not necessary for causality. A generalized free causal field theory carrying a reducible representation is described in
R. F. Streater,
Local fields with the wrong connection between spin and statistics,
Comm. Math. Phys. Volume 5, Number 2 (1967), 88-96.
Recent work by Schuster & Toro does not contradict the findings reported in the above references, as the latter are not constructing local and gauge invariant fields, hence fail to give their constructions a proper physical meaning. See this and this comment in this thread.
For possible relations to dark matter see these papers by Schroer: arxiv.org/abs1601.02477,