I wouldn't have guessed that anything that changed with a machine upgrade would affect an app's running.
I got the same pop-up when I launched TTP on my 2015 machine as I had previously gotten on my 2010 machine, and the only reason I even tried on either is that it's been posted here (by you, I think) that it was really only the OS X testing portions of TTP that were in question but that the disk testing portions were OK to run.
My use of the term
upgrade apparently caused confusion. I was thinking in terms of software upgrades
ie. going from OS X 10.10 (Yosemite) to OS X 10.11 (El Capitan)
not hardware upgrades. My use of the term
update was intended to refer to software updates,
ie. OS X 10.11.3 TO 10.11.4.
It is not uncommon for Apple to change the arguments (options) of commands and APIs, change what they do, or even eliminate them altogether as part of an OS upgrade. Some changes may simply break the app and cause it not to work, others may actually result in unintended damage to the system. Such changes are announced well in advance but not all developers keep up with the changes and even those that do may find they have misconstrued the effect of a change, so careful testing is essential for safety.
Unless an app such as Diskwarrior, TechTool Pro, Drive Genius, MacPilot, OnyX, Tinkertool, Cocktail, TinkerTool System,
etc. actually writes to the drive it is extremely unlikely to do any harm, but that does not guarantee an out of date app will run under a newer OS than it was certified for.
