We have performed scanning photocurrent microscopy measurements of field-effect transistors (FETs) made from individual ultraclean suspended carbon nanotubes (CNTs). We investigate the spatial-dependence, polarization-dependence, and gate-dependence of photocurrent and photovoltage in this system. While previous studies of surface-bound CNT FET devices have identified the photovoltaic effect as the primary mechanism of photocurrent generation, our measurements show that photothermoelectric phenomena play a critical role in the optoelectronic properties of suspended CNT FETs. We have quantified the photothermoelectric mechanisms and identified regimes where they overwhelm the photovoltaic mechanism.