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The Traveling Salesman Problem (TSP) is a well-known combinatorial optimization problem. We are concerned here with online versions of a generalization of the TSP on metric spaces where the server doesn't have to accept all requests. Associated with each request (to visit a point in the metric space) is a penalty (incurred if the request is rejected). Requests are revealed over time to a server, initially at a given origin, who must decide which requests to serve in order to minimize the time to serve all accepted requests plus the sum of the penalties associated with the rejected requests. In a first online version of this problem (basic version), we assume that the server's decision to accept or reject a request can be made any time after its release date. In a second online version of this problem (real-time version), we assume that the server's decision to accept or reject a request must be made exactly at its release date. After reviewing prior results on the online TSP, we first provide an optimal 2-competitive online algorithm for the basic version of the problem in a general metric space, improving prior results from the literature. We then consider the real-time version of the problem and show that there can't be any finite $c$-competitive online algorithm in a general metric space.