In the Internet of Things (IoT) environment, edge computing can be initiated at anytime and anywhere. However, in an IoT environment, edge computing sessions are often ephemeral, i.e., they last for a short period of time and can often be discontinued once the current application usage is completed or the edge devices leave the system due to factors such as mobility. Therefore, in this paper, the problem of ephemeral edge computing in an IoT is studied by considering scenarios in which edge computing operates within a limited time period. To this end, a novel online framework is proposed in which a source edge node offloads its computing tasks from sensors within an area to neighboring edge nodes for distributed task computing, within the limited period of time of an ephemeral edge computing system. The online nature of the framework allows the edge nodes to optimize their task allocation and decide on which neighbors to use for task processing, even when the tasks are revealed to the source edge node in an online manner, and the information on future task arrivals is unknown. The proposed framework essentially maximizes the number of computed tasks by jointly considering the communication and computation latency. To solve the joint optimization, an online greedy algorithm is proposed and solved by using the primal-dual approach. Since the primal problem provides an upper bound of the original dual problem, the competitive ratio of the online approach is analytically derived as a function of the task sizes and the data rates of the edge nodes. Simulation results show that the proposed online algorithm can achieve a near-optimal task allocation with an optimality gap that is no higher than 7.1% compared to the offline, optimal solution with complete knowledge of all tasks.