Analytical calculations and Monte Carlo simulations of a lattice model of a tetravalent network-forming fluid show liquid-liquid immiscibility at low temperatures. Three types of phase behavior are found to occur : Vapor-liquid equilibrium, vapor-liquid equilibrium plus liquid-liquid immiscibility with an upper critical temperature, and vapor-liquid equilibrium plus closed-loop liquid-liquid immiscibility. The coexisting liquids differ appreciably in density, structure, and molecular dynamics. The low-density liquid displays more structured, open networks, and slower translational and network-restructuring dynamics than the high-density Liquid. The model provides useful insights into the molecular basis of low-temperature immiscibility in network-forming fluids, and its relation to the unusual transition between distinct amorphous phases in vitreous water.