OK, wheels up in my head turning again
If that was intentional airplane jargon, and that was the best you could come up with, then you should be PUN-ished.
(Kidding, also failing at trying to be funny.)
Will I need to be sure that none of the circuitry that is connected to the leads of the DUT in anyway touches the case?
In general, you should make the circuit board as covered with insulation as you can get it. It would be really bad for anything conductive to touch any part of the circuit board at all.
Speaking specifically, if our load circuit has it's ground attached to the case ground, then anything from our circuit that touches the case will be grounded. This could be as simple as screwing with the feedback causing things to flake out. Or of course it could just let a whole lot of current through one single part, instantly exploding it. And I really do mean exploding, like, with a loud bang and smoke and such.
Things probably will be hooked up this way. So short answer is yes, keep anything electronic from touching the case unless specifically told to do otherwise.
[Would] the second PSU in the 24v series set up no longer be isolated from ground if anything touches the now grounded case?
No, that supply is still "isolated" in the way it needs to be. It is the high side, the most positive supply. And it's positive is the only wire coming out of it to the rest of the world really. The rest is isolated. The thing that needed to be accounted for so that things were safe was the other supply's positive. Which is why you needed to isolate the other one's ground, because that ground and supply were attached directly to each other.
Basically, The statement...
I have to isolate one of the PSUs from ground or it will short when I connect them in series
Should be corrected to...
I have to isolate one of the PSU's Grounds from it's case otherwise it will short the other supply when I connect the two together