Why would the elimination of the value of labor result in poverty and inequality? It should be the opposite, as poverty and inequality is the current status quo (for the many).
Because labor is the only thing the working class can leverage against the capitalists. They sell their labor for wages to the owners who have the means of production and capital. If the working class can't bargain its labor anymore, it ceases being useful/tolerated by the bourgeoisie (who owns everything, including the state and police). See the issue now?
This isn't theory, ask the Luddites why they got so mad when their employers started buying machines to replace them. They didn't get richer and freer: they were thrown out to rot on the pavement, while their ex-employers kept 100% of the productivity increases.