The linear response is studied in globally coupled oscillator systems including the Kuramoto model. We develop a linear response theory which can be applied to systems whose coupling functions are generic. Based on the theory, we examine the role of asymmetry introduced to the natural frequency distribution, the coupling function, or the coupling constants. A remarkable difference appears in coexistence of the divergence of susceptibility at the critical point and a nonzero phase gap between the order parameter and the applied external force. The coexistence is not allowed by the asymmetry in the natural frequency distribution but can be realized by the other two types of asymmetry. This theoretical prediction and the coupling-constant dependence of the susceptibility are numerically verified by performing simulations in $N$-body systems and in reduced systems obtained with the aid of the Ott-Antonsen ansatz.