The 'action at a distance' referred to here it not the one that crops up in entanglement. In entanglement, the nature of our knowledge changes instantaneously when we make a measurement. The action at a distance problem Weinberg refers to relates to the speed at which one object influences a distant one. For example, in Newtonian gravity, with its V = GM1M2/r potential, the speed of propagation of gravity is effectively infinite.