Wikipedia is correct. The derivation of Lehmann, Symanzik and Zimmermann is invalid if the theory contains long-range forces (massless gauge bosons). As pointed out by Nihar in the comments this is known as the infrared problem of QFT. It is (as far as I know) a still unsolved problem in the branch of mathematical physics that tries to construct a mathematically rigorous formulation of QFT. The best rigorous and at the same time (semi-)pedagogical treatment that I know of this topic can be found in O. Steinmann, "Perturbative Quantum Electrodynamics and Axiomatic Field Theory", https://link.springer.com/book/10.1007/978-3-662-04297-7, Chapter 6.3.
However, particle physics has a long and proud tradition of cutting corners on mathematical rigour and still being able to make correct predictions. Let's suspend our concerns for a moment and just apply to LSZ formulat to, say, QED. We then start calculating some scattering amplitudes and we eventually find that our amplitudes are divergent in some kinematic regions. These divergences are a sign that we've made a mistake: we've used the LSZ formula in a situation where it is not valid. We don't know how to fix the mistake (there's no version of the LSZ formula which holds for gauge theories), but we can deal with the divergences in a way that's very similar to the way we deal with ultraviolet (UV) divergences: we regularise them and cancel them out.
In QED the regularisation is rather straightforward. We can simply add a photon mass $\lambda$ to the theory. This means that for $\lambda > 0$ the LSZ formula actually applies. The IR divergences manifest as terms which diverge when $\lambda$ goes to zero.
To cancel the divergences you have to calculate sufficiently 'inclusive' cross sections. For example, one situation where you encounter an IR divergence is when you emit a very 'soft' (i.e. low energy) photon. Real detectors need a minimum amount of energy before they can register the photon, so if your detectors sees no extra photon that could mean there was no extra photon or there was one but it was too soft to register. So you have to combine your cross sections for processes with a soft photon in the final state with the cross sections for processes with no soft photon in the final state since you can't distinguish the two. The IR divergences cancel in the sum.