Ok so less than 30 minutes ago my Note 3 was working fine! I put it in my pocket and looked at it again and it was off and wouldn't turn on. It had plenty of battery left at the time though I don't know exactly how much. First thing I did was take it to the charger. The battery icon wouldn't show up so I pulled the battery and tried booting into recovery. That did not work either. I tried to power it on without the battery in via the AC charger. (Yes some phones will boot without a battery). That also failed. I tried a different AC adapter of the same rating. It failed. I tried a different cable, fail. Etc etc. Any ideas? Also it's worth noting that when it was plugged up to the charger it got reasonably warm. The battery didn't get warm. The area around the Snapdragon 800 got warm. With and without the battery in whilst on the charger it still got very warm. (Like it was trying to do something very load intensive?)
It sounds like a defective unit. I would send it in for warranty replacement. If it is getting warm while plugged in the power is shorting out the unit causing it to show no life but receive power.