The tunneling heterojunctions made of two-dimensional (2D) materials have been explored to have many intriguing properties, such as ultrahigh rectification and on/off ratio, superior photoresponsivity, and improved photoresponse speed, showing great potential in achieving multifunctional and high-performance electronic and optoelectronic devices. Here, we report a systematic study of the tunneling heterojunctions consisting of 2D tellurium (Te) and Tin disulfide (SnS2). The Te/SnS2 heterojunctions possess type-II band alignment and can transfer to type-III one under reverse bias, showing a reverse rectification ratio of about 5000 and a current on/off ratio over 104. The tunneling heterojunctions as photodetectors exhibit an ultrahigh photoresponsivity of 50.5 A W−1 in the visible range, along with a dramatically enhanced photoresponse speed. Furthermore, due to the reasonable type-II band alignment and negligible band bending at the interface, Te/SnS2 heterojunctions at zero bias exhibit excellent self-powered performance with a high responsivity of 2.21 A W−1 and external quantum efficiency of 678%. The proposed heterostructure in this work provides a useful guideline for the rational design of a high-performance self-powered photodetector.