Photodiodes are ubiquitous in industry and consumer electronics. New applications for photodiodes are constantly emerging, such as the internet of things and wearable electronics that demand different mechanical and optoelectronic properties from those provided by conventional inorganic devices. This has stimulated considerable interest in the use of next generation semiconductors, particularly the organics, which provide a vast palette of available optoelectronic properties, can be incorporated into flexible form factor geometries, and promise extremely low cost, low embodied energy manufacturing from earth abundant materials. The sensitivity of a photodiode to low light intensities (typically important in these new applications) depends critically on the dark current. Organic photodiodes, however, are characterized by a much higher dark current than expected for thermally excited band-to-band transitions. Here, we show that the lower limit of the dark current is given by recombination via mid-gap trap states. This new insight is generated from temperature dependent dark current measurements of narrow-gap photodiodes for the near-infrared. Based on Shockley-Read-Hall statistics, a diode equation is derived which can be used to determine an upper limit for the specific detectivity and to explain the general trend observed for the light to dark current ratio as a function of the experimental open-circuit voltage for a series of organic photodiodes. A detailed understanding of the origins of noise in any detector is fundamental to defining performance limitations and thus is critical to materials and device selection, design and optimisation for all applications. Our work establishes these important principles for organic semiconductor photodiodes for the near-infrared.