To enable sustainable power generation and transportation, green hydrogen is currently envisioned as a future fuel for combustion systems. However, combustion of hydrogen differs significantly from commonly used fossil fuels spanning a larger range of spatial scales. To overcome these challenges, adaptive mesh refinement (AMR) is an effective approach, enabling full spatial resolution of the physical phenomenon of interest at a drastically reduced computational cost. Efficient load balancing is also required for multi-node simulations of these systems. Two particularly challenging configurations that could benefit from load-balanced AMR are considered: lean hydrogen/air flames exhibiting thermo-diffusive instabilities and under-expanded hydrogen jets. In these systems, resolving the thin and transient flame front and the high-density gradients leads to severe imbalances between processor loads, making load-balancing mandatory. In this work, we develop an approach for efficient dynamic load-balancing in combination with AMR and demonstrate its application to the two configurations mentioned. Performance characteristics are discussed. Furthermore, the configurations are known to be sensitive to mesh resolution. Therefore, the influence of AMR and cell-type choice between hexahedrons and polyhedrons on the small-scale structures are discussed. The implementations and numerical solutions are carried out in the open-source framework OpenFOAM, extending the existing AMR of hexagonal meshes to general polyhedral meshes and introducing load-balancing capabilities for both mesh types. Acknowledgment The authors gratefully acknowledge the NHR Center NHR4CES at TU Darmstadt for supporting this work and providing the needed computing time. This is funded by the Federal Ministry of Education and Research, and the state governments participating on the basis of the resolutions of the GWK for national high performance computing at universities (www.nhr-verein.de/unsere-partner). This research was part of the "Center of Excellence in Combustion", which received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement Noº 952181.