Does the photoelectric effect obey Ohm's law?

does this mean that Ohm's law just fails in this case

Ohm's law is not universal. The ideal resistor circuit element is defined by Ohm's law but not all circuit elements obey Ohm's law; Ohm's law only applies to ohmic devices.

Physical resistors and conductors approximately obey Ohm's law but, for example, semiconductor diodes, transistors, thyristors, solar cells, vacuum tubes, batteries, thermistors, etc. etc. etc. do not.

The term Resistance does not come into play while dealing with Photoelectric Effect. The latter is related to the emission of electrons when the surface of a metal(or any substance) is hit by photon particles(photon is the unit particle making up the light that we talk of). Here the more important concept is that of Work Function, i.e. the minimum amount of energy required to release electrons from the surface of the substance. Current does not flow inside the metal or the given substance. It is an interaction between the incident photon and the outermost electrons. Photo current is produced when the emitted photo electrons reach the oppositely charged plate. See:

However Ohm's Law comes into play when the electrons travel through the given substance. The electrical resistance of an electrical conductor is the opposition to the passage of an electric current through that conductor. It depends upon the length and area of cross- section of the conductor. This is concerned with the motion of electrons inside the conductor. The current generated depends on drift velocity. Check out: