Climate warming is subjecting plants to heat stress, which can affect their physiological processes thereby impacting their growth, development, and productivity. Potato (Solanum tuberosum L.) is a staple food worldwide, but potato crops are very sensitive to heat stress. We have studied the effects of heat stress on the leaf chlorophyll content, plant growth, and tuber yield of 55 commercial potato cultivars in clonal tests under heat-stress conditions [HS; 35 °C (day), 28 °C (night)] and control (non-stress) conditions [CK; 22 °C (day), 18 °C (night)]. The potato cultivars varied in their response to heat stress. Overall, heat stress reduced leaf size, increased the SPAD index values for leaf chlorophyll by up to 65%, and increased plant height by 64%, but severely reduced (by 93%) the mass of the largest tuber. The HS:CK SPAD ratios positively correlated with the HS:CK plant height ratio, mass of the largest tuber under heat stress, and the HS:CK ratio for mass of the largest tuber. Potato cultivars displayed a correlated response to heat stress for their leaf chlorophyll content, plant height, and tuber mass. We have identified the most heat-tolerant and heat-susceptible cultivars for these traits. Under heat-stress conditions, potato cultivars tend not to show as much reduction in tuber mass if the plants have greater increases in leaf chlorophyll content and plant height.