This paper proposes an optimization-based task and motion planning framework, named "Logic Network Flow," that integrates temporal logic specifications into mixed-integer programs for efficient robot planning. Inspired by the Graph-of-Convex-Sets formulation, temporal predicates are encoded as polyhedral constraints on each edge of a network flow model, instead of as constraints between nodes in traditional Logic Tree formulations. We further propose a network-flow-based Fourier-Motzkin elimination procedure that removes continuous flow variables while preserving convex relaxation tightness, leading to provably tighter convex relaxations and fewer constraints than Logic Tree formulations. For temporal logic motion planning with piecewise-affine dynamical systems, comprehensive experiments across vehicle routing, multi-robot coordination, and temporal logic control on dynamical systems using point-mass and linear inverted pendulum models demonstrate computational speedups of up to several orders of magnitude, alongside reduced memory consumption. Hardware demonstrations with quadrupedal robots validate real-time replanning capabilities under dynamically changing environmental conditions. The project website is at https://logicnetworkflow.github.io/.

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