Bionanohybrids of carbon nanotubes and fluorescent proteins (FPs) are a promising class of materials for optoelectronic applications. Understanding and controlling the charge transport mechanism between FPs and carbon nanotubes are critical to achieving functional reproducibility and exploring novel synergetic effects. This work demonstrates a novel phenomenon of photocurrent generation in field-effect transistors based on the conjugation of an individual single-walled carbon nanotube (SWCNT) and FPs. When studying the effect of gate voltage on the photoresponse, reversible switching from fast positive to a slow negative photoresponse in bionanohybrids associated with depletion and accumulation modes, respectively is observed. The latter demonstrates a stable memory effect after the light is turned off. It is revealed that in depletion mode, the charge carriers from the protein are not trapped at the interface due to effective screening by the gate potential. It is suggested that the main mechanism in photoresponse switching is a competitive effect between photogating and effective photodoping of the SWCNT by charges trapped at the nanotube interface. The noticeable effect of water molecules can support proton transfer as the main mechanism of charge transfer. This result illustrates that SWCNT/FP bionanohybrids bear great potential for the realization of novel optoelectronic devices.