Nonlinear optical responses, such as shift current and second-harmonic generation (SHG), of topological semimetals have been subjects of great interest because the momentum-space topological structures near Weyl or Dirac points lead to unusually large responses. This is especially important for technological applications as the shift current and SHG are often used for solar cell and frequency-changing applications for lasers, respectively. We demonstrate that nodal line semimetals can offer even larger responses at low frequencies. For example, we show that a large SHG response arises at finite doping when an external dc electric field is applied to break the inversion symmetry of an otherwise inversion-symmetric model of a nodal line semimetal. Furthermore, we introduce a model with intrinsically broken inversion symmetry, where both the shift current and SHG are singular in a range of parameters.