My working 533DM has a type 83 tube along with the type 5Y3 tube. The 816 is quite different. Hickok may have made changes in the circuit over time or someone may have modified the tube tester at some time but that tube is where I would start. First check the filament voltage where the 816 tube is inserted. If it is 2.5-volts, then the 816 may be correct. If it is 5-volts, it most likely should be a type 83 tube (if circuit changes were made to accommodate the 816 tube, they will need to be reversed). Second check for plate voltage. The 83 tube has two plates both of which should have high AC voltage on them when the P4 button is pressed (possibly also when not pressed). The 816 only has one plate which should have high AC voltage on it when the P4 button is pressed. An open filament on either type rectifier tube will also cause the problem being experienced.
I encountered the same symptom and found it to be a broken transformer winding at the lug for the plate of the 83 tube.
Oops, yes it was an 83 not an 816. Sorry and thanks all for your input.