Hello Jim,
I'll try to break down what I know, and PJ can come along and fill in the cracks (or chasms as they sometimes are).
The TL431 operates in our circuit by receiving its operating voltage from the cathode of the tube operating in the shunt regulator circuit. Therefore, it is important to consider the operating limits of the TL431, in that it must not be presented more than 32v and no less than 4-ish v. Also, the TL431 is limited to passing 100ma.
Next is to consider the circuit that follows after the shunt regulator. For anything containing a cathode follower, the years of experience and wisdom (before my time) suggest to pass at least as much current through the shunt reg tube as passes through the signal circuit itself. Otherwise, you can pass considerably less current through the shunt regulator and obtain a similar benefit.
So, considering those two points, you will know about how much current you want to run through your shunt regulator, and you will know your regulated voltage, so you can use Ohm's Law to calculate the approximate plate dissipation seen at the shunt reg tube. This will rapidly eliminate plenty of tubes out there as possible candidates (mostly those lightweight high gain triodes).
Since you know the approximate current you'd like to run through the shunt regulator, you can then look at triode curves for the tubes you have in mind and see what kind of cathode voltages you would expect under your operating conditions. I find that good tubes for these duties are low gain triodes, or reasonably beefy triode strapped pentodes. In my personal projects, I have found that the 6S4 works for just about everything I need, though it is seriously overkill at times.
Here is an example breakdown, of say a 6J5 transformer coupled preamp:
6J5 operating point - 4ma plate current, 2v bias, ~75v on the plate
Desired shunt regulator (per channel) - 150V output, 4ma for the 6J5, 2ma for the LED string on the 6J5 C4S, and perhaps 4-8ma of shunt regulator current (10-14ma total). Therefore, we would want to look at tubes that can take 0.008A*150v=1.2 watts. For the 6S4, with 150 volts on the plate and 8ma of current, the cathode voltage is around 6v, which is great. The plate dissipation is 8.5 watts, so it will be snoozing while your gear is running.
The last important consideration is the heat generated in the CCS feeding the shunt regulator and the rest of the circuit. This is approximately the difference between the regulated and unregulated voltages times the current, with some safety factor. With a 210V B+ supply feeding this regulator (IIRC, that is what the extended FP-III features), this would be a difference of 60V, and with total current set to 12ma, the MJE-5731A on the C4S board could be looking at about 0.75 watts, which is within the safe zone for that transistor.
There's also the feedback stabilization RC network hanging off the output of the shunt reg, PJ will have to comment on that one for sure!
-PB