The photovoltage produced by local illumination at the Schottky contacts of carbon nanotube field-effect transistors varies substantially with gate voltage. This is particularly pronounced in ambipolar nanotube transistors where the photovoltage switches sign as the device changes from p-type to n-type. The detailed transition through the insulating state can be recorded by mapping the open-circuit photovoltage as a function of excitation position. These photovoltage images show that the band-bending length can grow to many microns when the device is depleted. In our palladium-contacted devices, the Schottky barrier for electrons is much higher than that for holes, explaining the higher p-type current in the transistor. The depletion width is 1.5 mum near the n-type threshold and smaller than our resolution of 400 nm near the p-type threshold. Internal photoemission from the metal contact to the carbon nanotube and thermally assisted tunneling through the Schottky barrier are observed in addition to the photocurrent that is generated inside the carbon nanotube.