So we know that we can get some open circuit voltage by heating the cathode, but can it do any work? 

Here, two 6v3a diodes in series light an LED with their filaments at full tilt. The LED is drawing 7.5uA at 1.7v (LEDs are incredible)

Rough napkin math puts these diodes at about a 0.00012% conversion efficiency. That's between 5 and 6 orders of magnitude worse than a commercial PV cell. That said, these diodes are obviously not designed as thermionic converters. 

As a demonstration, it feels a little "so what" until you unplug the power and the LED keeps shining until the cathodes cool down. That's neat.