Actually there is an easy - well easy-ish - fix for this. OK it works for me, but I can't guarantee it'll work for you but there's no harm in trying.
Take out the battery. Now with the battery out, hold the power button down continuously for 30 seconds. Then put the battery back in and try to turn on the phone. If the battery is dead, you can get an Anker-brand battery charger along with two after market replacement batteries for a very affordable price of $25 on Amazon.
This is draining any residual power out of the phone's circuit board. It's essentially a hard reset. I work in a computer repair shop and we do this with laptops all the time. Works about 50-60 percent of the time when the laptop wont power up.
Note: AT&T and CPR Cell Phone Repair have both said this is probably a symptom of a bad battery, but I have observed my phone do this with my stock battery AND the brand new aftermarket ones, leading me to believe this problem has to do with a combination of the new Android 5.0 and/or the phone's firmware and the charging circuit of the phone. Could also be a CMOS and/or RAM issue. The phone charges if it's on but not if it is off. It does not matter how much charge the battery has, if I turn it off it will not turn on again w/out hard resetting it.
PS: It also does this with a custom recovery (TWRP) and a custom ROM (CloudyG3) so if it's a OS issue, it's with Android 5.0 in general. But It's probably a firmware issue.
OK....Weird....now, if i put the battery back in after resetting, it is still not powering up. But if I plug it in to the charger with the battery in the screen come up and says 75% charge. If I press the power button, it turns on. OK wtf???
If I restart instead of power off and then back on it works just fine. Totally mystified as to what could be going on here......