Many tobacco (Nicotiana tabacum) cultivars are salt-tolerant and thus are potential model plants to study the mechanisms of salt stress tolerance. The CALCINEURIN B-LIKE PROTEIN (CBL) is a vital family of plant calcium sensor proteins that can transmit Ca2+ signals triggered by environmental stimuli including salt stress. Therefore, assessing the potential of NtCBL for genetic improvement of salt stress is valuable. In our studies on NtCBL members, constitutive overexpression of NtCBL5A was found to cause salt supersensitivity with necrotic lesions on leaves. NtCBL5A-overexpressing (OE) leaves tended to curl and accumulated high levels of reactive oxygen species (ROS) under salt stress. The supersensitivity of NtCBL5A-OE leaves was specifically induced by Na+, but not by Cl−, osmotic stress, or drought stress. Ion content measurements indicated that NtCBL5A-OE leaves showed sensitivity to the Na+ accumulation levels that wild-type leaves could tolerate. Furthermore, transcriptome profiling showed that many immune response-related genes are significantly upregulated and photosynthetic machinery-related genes are significantly downregulated in salt-stressed NtCBL5A-OE leaves. In addition, the expression of several cation homeostasis-related genes was also affected in salt-stressed NtCBL5A-OE leaves. In conclusion, the constitutive overexpression of NtCBL5A interferes with the normal salt stress response of tobacco plants and leads to Na+-dependent leaf necrosis by enhancing the sensitivity of transgenic leaves to Na+. This Na+ sensitivity of NtCBL5A-OE leaves might result from the abnormal Na+ compartmentalization, plant photosynthesis, and plant immune response triggered by the constitutive overexpression of NtCBL5A. Identifying genes and pathways involved in this unusual salt stress response can provide new insights into the salt stress response of tobacco plants.