Chronic stress deteriorates the immune function of fish, thereby increasing their vulnerability to infections. However, the molecular and cellular mechanisms underlying stress-mediated immunosuppression and infection susceptibility in fish remain largely unknown. Understanding these mechanisms will contribute to improving fish welfare and their farm production. Herein, we review the challenges of sturgeon aquaculture in subtropical countries, where current climate change has giving rise to significant temperature increments during summer. This leads to the exposure of fish to stressful conditions during these months. Chronic heat stress deserves attention considering the rapid warming rate of the planet. It is already affecting wild fish populations, with disastrous consequences for sturgeons, which are one of the most endangered fish species in the world. In this context, we discuss the most recent advances through the studies on the effects of chronic heat stress on the innate immune components of sturgeons. To this end, we summarise the findings of studies focusing on the aquaculture of Russian sturgeons and observations made on other Acipenser species. Special attention is given to acute-phase proteins, as they might be valuable biomarkers of heat stress and infection, with applicability in monitoring the fish health status in farms.