Most cultivated potatoes are tetraploid, and the tuber is the main economic part that is consumed due to its calorific and nutritional values. Recent trends in climate change led to the frequent occurrence of heat and drought stress in major potato-growing regions worldwide. The optimum temperature for tuber production is 15–20 °C. High-temperature and water-deficient conditions during the growing season result in several morphological, physiological, biochemical, and molecular alterations. The morphological changes under stress conditions may affect the process of stolon formation, tuberization, and bulking, ultimately affecting the tuber yield. This condition also affects the physiological responses, including an imbalance in the allocation of photoassimilates, respiration, water use efficiency, transpiration, carbon partitioning, and the source–sink relationship. The biochemical responses under stress conditions involve maintaining ionic homeostasis, synthesizing heat shock proteins, achieving osmolyte balance, and generating reactive oxygen species, ultimately affecting various biochemical pathways. Different networks that include both gene regulation and transcription factors are involved at the molecular level due to the combination of hot and water-deficient conditions. This article attempts to present an integrative content of physio-biochemical and molecular responses under the combined effects of heat and drought, prominent factors in climate change. Taking into account all of these aspects and responses, there is an immediate need for comprehensive screening of germplasm and the application of appropriate approaches and tactics to produce potato cultivars that perform well under drought and in heat-affected areas.