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Robust power allocation for OFDM wireless network localization is critical for accurate user localization. The goal is to minimize total transmission power while achieving a certain positioning accuracy despite uncertainties in channels and agent positions. A framework for robust power allocation in OFDM wireless network localization is presented, using the squared position error bound (SPEB) as the fundamental limit of localization accuracy. The SPEB is derived for OFDM signals in a synchronous network, considering deterministic unknown agent positions and channels, as well as a collection of agent positions and channels drawn from suitable distributions. A method is developed to obtain subcarrier power allocations that minimize total transmitted power, subject to a constraint on the average SPEB, for both unicast and multicast transmissions. Simulation results show that robust power allocation improves localization accuracy compared to non-robust power allocation, with only a limited power penalty.