Heterogeneous multiprocessor system-on-chips (MPSoCs) are suitable platforms for real-time embedded applications that require powerful parallel processing capability as well as low power consumption. For such applications, soft-error reliability (SER) due to transient faults and lifetime reliability (LTR) due to permanent faults are both key concerns. There have been several efforts in the literature oriented toward related reliability problems. However, most existing techniques only concentrate on improving one of the two reliability metrics, which are not suitable for embedded systems deployed in critical applications in need of a long lifetime as well as a reliable execution. This article develops a novel heterogeneous earliest-finish-time (HEFT)-based algorithm to maximize SER and LTR simultaneously under the real-time constraint for dependent tasks executing on heterogeneous MPSoC systems. More specifically, a new deadline-constrained reliability-aware HEFT algorithm, namely DRHEFT, is proposed, which seeks for the best SER–LTR tradeoff solutions through using fuzzy dominance to evaluate the relative fitness values of candidate solutions. The extensive experiments on real-life benchmarks as well as synthetic applications demonstrate that DRHEFT is capable of achieving better SER–LTR tradeoff solutions with higher hypervolume and less computation cost when compared with the state-of-the-art approaches.