You won't find such a requirement in the NEC. By way of illustration, for kitchen range hoods, the requirement is spelled out in NFPA standards 17A and 96.
From a technical viewpoint, the trigger mechanisms for nearly all but the smallest systems have microswitches at the trip mechanism. It's probably just as easy to ti in here, than to have a separate push button.
I couldn't think of anything like that in NEC, and I couldn't find anything in IFC either. I figured that if there was a requirement, someone here would certainly know it off the top of their head and save me hours of searching! It's unfortunately far easier to find an applicable requirement than to verify that there is no requirement.
NFPA-13 8.5.10 requires sprinklers in electrical equipment rooms (but all the equipment can be NEMA3, which makes this seem rather pointless), and 220.127.116.11 requires IT areas be valved separately from other areas. I found no other requirements in NFPA 13.
NFPA 76 6.5 requires a means to disconnect power be identified, but not that it be automatic. 6.6.1 requires ventilation be automatically dampered when smoke is detected. (Not upon sprinkler activation.) 10.6 requires that depowering occur only after the investigation determines that depowering will mitigate the event.
Well, THAT surprised me! I didn't find a requirement that says I have to put a shunt trip on a data center, but I *did* find a piece of code that specifically prohibits it!
Electronic equipment installations must be located in buildings protected by wet-pipe automatic sprinklers. Provide complete coverage throughout the building including electronic equipment areas. Protect electrical equipment installations by disconnecting the power upon activation of the fire protection system.
The computer room disconnect is in 645.10 It has to close the HVAC dampers but it doesn't need to be activated by the fire suppression system. The controls need to be grouped at "the principle exit doors". Usually you have a halon trip, a halon abort and an EPO all under plastic bubbles. The sequence for the halon is a klaxon goes off and x number of seconds later the halon dumps. You have those seconds to decide if this is really a fire or if you want to save the $100,000 worth of halon (the 1996 price, it is probably $300k now) for another day. I have been in a halon dump, no fun. (think tornado) I also cleaned up after another one. They might use dry pipe sprinklers in computer rooms now but I never saw it much in the 20th century. Wet pipe is definitely not recommended. (leaks) AEC/DOE used to have a CO2 system in their Germantown Md computer room. There was some talk that OSHA made then pull it out.
I am surprised that DoD wants shunt trips on "mission critical systems". When I was in the military, mission critical systems had "battle override" that shunted out overcurrent devices. The thought was we will keep fighting until this thing burns up.
Yeah, there's always contention over EPO buttons, too! Surprise-surprise that operators don't like a giant red "push here to kill the mission" button right out in the open where anyone can push it. There's enough weasel words in 645.4 that they frequently get their way, too. Some might consider the lack of fire-rated walls a violation... others, the excuse they need to ignore other codes they don't like.
DoD instructions, MIL-HDBKs and the like are like NFPA, though, in that they're living documents that can be changed; easier, really, because there's no set time limit, they can be revised at any time, and there's usually only one person that needs convincing and an email from them is just as good as an official revision. It's just a matter of finding the person (or office) who's responsible for it and giving them a call.
The "anyone can push it" concern is why these things are usually under a bubble cover. In a big data center an unscheduled power off can cost $100,000 in data damage, lost productivity and lost revenue. It might take over an hour to get things going again and longer than that to recover any lost data. Of course some may be lost forever.