Production of heterologous proteins, especially biopharmaceuticals and industrial enzymes, in living cell factories consumes cellular resources. Such resources are reallocated from normal cellular processes toward production of the heterologous protein that is often of no benefit to the host cell. This competition for resources is a burden to host cells, has a negative impact on cell fitness, and may consequently trigger stress responses. Importantly, this often causes a reduction in final protein titers. Engineering strategies to generate more burden resilient production strains offer sustainable opportunities to increase production and profitability for this growing billion-dollar global industry. We review recently reported impacts of burden derived from resource competition in two commonly used protein-producing yeast cell factories: Saccharomyces cerevisiae and Komagataella phaffii (syn. Pichia pastoris). We dissect possible sources of burden in these organisms, from aspects related to genetic engineering to protein translation and export of soluble protein. We also summarize advances as well as challenges for cell factory design to mitigate burden and increase overall heterologous protein production from metabolic engineering, systems biology, and synthetic biology perspectives. Lastly, future profiling and engineering strategies are highlighted that may lead to constructing robust burden-resistant cell factories. This includes incorporation of systems-level data into mathematical models for rational design and engineering dynamical regulation circuits in production strains.