There is nothing indicating that Harry's life was protected by anything out of the ordinary. That's not to say he wasn't protected - Dumbledore took a keen interest in him and the rest of the staff at Hogwarts worked hard to keep all the students safe. It's notable that Hogwarts hadn't had a student die for decades, yet it bordered (and occasionally sent first years INTO) a forest filled with giant spiders, centaurs, etc. It had a tree on the grounds which could and would pummel anything in reach into submission, with no indication that it would stop when the subject ceased moving. The staircases within the school similarly posed normal dangers - they frequently moved with students on them, would have steps disappear, etc.
The students, therefore, were under some serious protection just to stay alive at first.
That said, there's no possible way that Harry could have died before facing Lord Voldemort.
From an out of Universe perspective, it would make for a terrible book.
From an in-Universe perspective, neither Harry nor Voldemort could die except by the other's hand. Prophecies, in the HP Universe, cannot be wrong. They can be misread, misconstrued, misrepresented, or otherwise turn out entirely unlike they were expected to, but they will always make sense, once fulfilled, to the person who originally heard them.
The prophecy that foretold Voldemort's downfall (which could have applied to Neville, had Voldemort chosen to mark him instead) was clear on one point: neither Voldemort nor the child he marked would die except by the other's actions.
The prophecy states:
"The one with the power to vanquish the Dark Lord approaches ... born
to those who have thrice defied him, born as the seventh month dies
... and the Dark Lord will mark him as his equal, but he will have
power the Dark Lord knows not ... and either must die at the hand of
the other for neither can live while the other survives ... the one
with the power to vanquish the Dark Lord will be born as the seventh
month dies ..."
Ergo, it was certainly possible for Harry to die a natural death while a Horcrux, but it was already known that he wouldn't.