Full-duplex device-to-device (D2D) communications over cellular networks is a promising solution for maximizing wireless spectral efficiency. However, in practice, due to the unpredictable arrival of D2D users, the base station (BS) must smartly allocate suitable channels to arriving D2D pairs. In this paper, the problem of dynamic channel allocation is studied for full-duplex D2D networks. In particular, the goal of the proposed approach is to maximize the system sum-rate under complete uncertainty on the arrival process of D2D users. To solve this problem, a novel approach based on an online weighted bipartite matching is proposed. To find the desired solution of the channel allocation problem, a greedy online algorithm is developed to enable the BS to decide on the channel assignment for each D2D pair, without knowing any prior information on future D2D arrivals. For an illustrative case study, upper and lower bounds on the competitive ratio that compares the performance of the proposed online algorithm to that of an offline algorithm are derived. Simulation results show that the proposed online algorithm can achieve a near-optimal sum-rate with an optimality gap that is no higher than 8.3% compared to the offline, optimal solution that has complete knowledge of the system.