Intelligent reflecting surface (IRS) has shown its effectiveness in facilitating orthogonal time-division integrated sensing and communications (TD-ISAC), in which the sensing task and the communication task occupy orthogonal time-frequency resources, while the role of IRS in the more interesting scenarios of non-orthogonal ISAC (NO-ISAC) systems has so far remained unclear. In this paper, we consider an IRS-aided NO-ISAC system, where a distributed IRS is deployed to assist concurrent communication and location sensing for a blind-zone user, occupying non-orthogonal/overlapped time-frequency resources. We first propose a modified Cramer-Rao lower bound (CRLB) to characterize the performances of both communication and location sensing in a unified manner. We further derive the closed-form expressions of the modified CRLB in our considered NO-ISAC system, enabling us to identify the fundamental trade-off between the communication and location sensing performances. In addition, by exploiting the modified CRLB, we propose a joint active and passive beamforming design algorithm that achieves a good communication and location sensing trade-off. Through numerical results, we demonstrate the superiority of the IRS-aided NO-ISAC systems over the IRS-aided TD-ISAC systems, in terms of both communication and localization performances. Besides, it is shown that the IRS-aided NO-ISAC system with random communication signals can achieve comparable localization performance to the IRS-aided localization system with dedicated positioning reference signals. Moreover, we investigate the trade-off between communication performance and localization performance and show how the performance of the NO-ISAC system can be significantly boosted by increasing the number of the IRS elements.