Digital twin (DT) can help create a digital representation of a physical system, thereby reflecting its real-time status. The digital object, often called cyber twin (CT), facilitates real-time monitoring and control of the physical object, i.e., the so-called physical twin (PT). Owing to this ability, CTs can optimize the PTs and simulate their status, without interrupting the physical world. Given the various CT use cases, one can identify two distinct types of DT tasks: 1) update tasks for PT-CT synchronization and 2) inference tasks for obtaining real-time testing responses. The diverse real-time requirements for update/inference tasks raise the task scheduling problem that has been neglected in previous studies. In this article, the real-time DT task scheduling problem is investigated. In particular, a new approach for evaluating the performance of real-time scheduling of DT tasks is introduced considering the relationship between update/inference tasks and fairness among CTs. Moreover, offline and online DT task scheduling schemes are proposed with the goals of maximizing the DT freshness ratio and minimizing task rejections. In particular, the DT freshness ratio maximization problem is formulated as an offline task scheduling scheme. The proposed offline solution can significantly reduce the solution space without losing optimality. Furthermore, the scheduling policies for achieving the maximal DT freshness ratio are established using which an online scheduling algorithm is designed. Simulation results show that the proposed offline/online schemes increase the DT freshness ratio by at least 16% and 11%, respectively, compared to benchmarks. The results also show that the task rejection ratio of the proposed online algorithm is within 8% of the lower bound.