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I have done hundreds of these installs and have never seen this issue. I can watch movies, surf the web, do pretty much anything that keeps the phone “awake.” However, anytime I let the screen go blank/timeout/idle (whichever you call it), it turns itself off after a few minutes. I installed the battery last night and was able to surf reddit for a few hours, then upon letting the phone go to sleep it turned itself off after a few minutes. This is the repetitive series of events that occur each time. I have tried soft and hard resets. Whats even stranger is that I cannot turn the phone back on unless I plug it into the wall. I am not able to turn the phone on using the power button. The battery is fully charges as I am able to use it unplugged as long as I don’t let the phone go to “sleep.”
You should try a full DFU Restore to eliminate any possible software/iOS corruptions issue. If that doesn’t work, then you will need to start isolating the major subcomponents (screen, cameras, sensors buttons etc) one at a time until you either find the culprit or isolate it down to the logic board. If you are uncomfortable doing these steps, consider finding a repair shop to do the troubleshooting. A shop that also does micro-soldering can go deeper if the problem is not modular in nature.
I changed my iphone 6 battery by myself with an aftermarket one and it worked normally for 1 week then it developed the same issues as you described. I complained and got another battery from the seller. With the battery replacement , the phone has been back to normal again ever since.