The only problem that may cause is thet (due to the apps you're running) the phone may be using more power than it's drawing from the charger, so it's not really charging, it's just discharging more slowly. But if the % charge keeps increasing, even slowly, it'll eventually charge. (I've used cellphones while they're charging since the first Motorola bricks, and still do it with the latest Androids and never have a problem. If the charge % keeps going down, I shut the phone off for a couple of hours, until it's fully charged.) The charge going into the battery is the amount of charge the phone can draw from the charger (the Note 3 can draw 1200mA, the 4 should be about the same) minus the amount being used by the phone. If the phone is off, the full amount goes to charging the battery. If the phone is on, part of it doesn't, and charging to 100% takes longer. The phone itself (the parts other than the battery and the charging circuit) don't even know if you're charging the battery or not.
(In fact, other than to swap batteries, which I do at least once a month [I always have at least 2 batteries for a phone], I don't think my Note 3's been turned off for months, and I charge it whenever it falls to about 40%. Looking at it another way, if the phone's connected to a computer and you're transferring files or something, the phone's being used while it's being charged. It's designed to do that. Whether it's a file transfer app, the phone app or any other app, it's just an app running while the phone is charging. [Using the phone to make or take a call might use more than the charger is supplying, but as soon as you finish the call, the phone starts charging again.])
@Tech Adfdict:
You had a few defective phones. The GPS receiver shouldn't get hot during use, whether the phone is charging or not. (And, as far as the battery is concerned, the GPS receiver is just a resistor, the same as any other part of the phone - it's DC, so .the hardware draws in-phase [or, actually, no phase] current - CPU, wifi receiver, GPS receiver - it's a resistor.)