That is not true, any cylinder stamped with DOT is always required to be within hydro test when it is filled no matter what the use. Just because its tucked away in the corner forever does not exempt it.
We are not in complete disagreement, but you are being a bit too broad in your application of testing requirements. Generally speaking a "K" bottle is going to be considered portable regardless of where it is located by most DOT inspectors, so they tned to get requalified per DOT regs. A surplus submarine ballast tank is normally not portable and would not be testable by most facilities, so the odds of it ever being tested are slim and none.
From a strictly legal perspective it is simply a matter of jurisdiction. The DOT, formerly the Interstate Commerce Commission (ICC), only has jurisdiction over tanks that are used, or in some interpretations, could potentially be used, in interstate commerce. That is a distinctly different issue than whether periodic hydro testing is a good idea. So if the tank is not portable or is built in to the structure in such a way as to not be readily removeable from the building, the DOT regs are not going to be applicable. Consequently, they can technically be filled even without a current requalification stamp.
Also, due to the very small wall stress involved, tanks below a certain diameter and lenght do not require a hydro at all even if DOT stamped.
The grey area becomes user owned scuba and welding tanks that are not used in interstate commerce - but are portable and could eventually cross a state line or be used in commerce. DOT inspectors tend to assume the latter.
Local state, county and city regulations however may require compliance with the DOT regs or similar rules as a matter of local, not federal, jurisdiction.
You can adopt a holier than thou attitude about tank testing and look at it as a very black and white issue, but the DOT regs still won't apply in all cases.