Task-based programming models such as OpenMP 5.0 and OmpSs are simple to use and powerful enough to exploit task parallelism of applications over multicore, manycore and heterogeneous systems. However, their software-only runtimes introduce relevant overhead when targeting fine-grained tasks, resulting in performance losses. To overcome this drawback, we present a hardware runtime Picos++ that accelerates critical runtime functions such as task dependence analysis, nested task support, and heterogeneous task scheduling. As a proof-of-concept, the Picos++ hardware runtime has been integrated with a compiler infrastructure that supports parallel task-based programming models. A FPGA SoC running Linux OS has been used to implement the hardware accelerated part of Picos++, integrated with a heterogeneous system composed of 4 symmetric multiprocessor (SMP) cores and several hardware functional accelerators (HwAccs) for task execution. Results show significant improvements on energy and performance compared to state-of-the-art parallel software-only runtimes. With Picos++, applications can achieve up to 7.6x speedup and save up to 90 percent of energy, when using 4 threads and up to 4 HwAccs, and even reach a speedup of 16x over the software alternative when using 12 HwAccs and small tasks.