In this paper, we propose and study the robust facility leasing problem with penalties (RFLEP), generalizing several well-known problems including the facility leasing problem (FLE). In the RFLEP, we are given the locations of facilities and clients, along with an integer q. A facility can be leased to open at its location for a certain time interval it chooses and is available for connecting clients during its leasing time. Leasing a facility incurs a leasing cost. Each client arrives at a specific given time and could choose one of these three states when it arrives, which are to be connected to some leased facility and pay a connection cost, reject to be connected and pay a penalty cost, and decide to be an outlier. Each connection cost is related to the locations of the corresponding pair of facility and client, and assume that the connection costs are non-negative, symmetric, and satisfy the triangle inequality. The objective is to lease some facilities and decide the state of each client, such that the total number of outliers is at most q, and the total leasing, connection, and penalty cost is minimized. As our main contribution, we design a primal-dual 3-approximation algorithm for the RFLEP. One of the main difficulties in solving the RFLEP is that the integrality gap of its natural linear program relaxation is unbounded. To overcome this difficulty, we design our algorithm based on a modified linear program relaxation, which guarantees that some very expensive leasing costs will not be incurred during the dual ascent process and leads to a constant-factor approximation ratio.