Why is my phone refusing to stay on?

This sounds like it could be a couple off things, but a few spring to mind. If the phone is powering up okay, but then just shuts off (not reboots) this is pretty typical of it cannot sustain enough power to remain on.

This is typical of a charge port with debris in it. It gets the power on signal from the charger, but cannot transfer enough energy to charge the phone rapidly enough. Does your charger sit flush with the bottom of the phone? If not, that';s a tell tale sign. I would try and have a look, and clean out any debris with a toothpick or something non conductive.

Could also be that the battery is bad. Especially if you have never replaced it, the iPhone 6 has been around for a number of years now and i would surprised if your original battery was still in good health.

I might have a look at the iPhone Randomly Shuts Off page as well for some more detailed info.