We analyze the two-body momentum correlation function for a uniform weakly interacting 1D Bose gas. We show that the strong positive correlation between opposite momenta, expected in a Bose-Einstein condensate with a true long-range order, almost vanishes in a phase fluctuating quasicondensate where the long-range order is destroyed. Using the Luttinger liquid approach, we derive an analytic expression for the momentum correlation function in the quasicondensate regime, showing: (i) the reduction and broadening of the opposite-momentum correlations (compared to the singular behaviour in a true condensate), and (ii) an emergence of anti-correlations at small momenta. We also numerically investigate the momentum correlations in the crossover between the quasicondensate and the ideal Bose gas regimes using a classical field approach and show how the anti-correlations gradually disappear in the ideal gas limit.