That said, few people will pay nothing - what they won't be paying if they live off share return (be it dividends or capital gains) is national insurance. I'm assuming that's what the person you replied to meant.
However that means nothing in terms of NHS funding any more, as NI just goes into the general treasury fund in practice.
It's become part of a convenient fiction about UK tax rates, given that most people are unaware of the way NI is structured. In effect the "20%" marginal tax band is roughly 32% marginal when you factor in NI, and the "40%" marginal tax band is really 42%, and the 45% band is effectively 47%.
The 20/40/45 looks a lot less progressive when it's actually about 32/42/47 (even less so when you factor in the many tax breaks those of us on the higher or additional rate can afford to take advantage of that lower earners can't)