I always charge my device while being on so as not to miss any calls or SMS... in fact, the only time I take it out of the charger is when I have to leave. Nothing to worry about specially if your charger and batteries are original. They are meant to complement each other so as not to overcharge. Today's Lithium ion batteries can tolerate an "always charge" situation and would not like going down a certain level or a fully drained level compared to Nicad or even NiMetal Hydride batteries.

The only time I turn off the device while charging is when I want it to charge faster specially after a rare time of being fully drained.

It doesn't really matter if it's on or off. What matters is that you do not drain it too much or keep it fully charged for long periods of time.

Li-ion batteries should not be completely drained. But, it is a little known fact that they should also not be kept at 100% charge. Take your laptop battery for example, if you keep it plugged in, it will damage the battery and actually reduce its life. This is why most new laptops have a feature where the battery is kept at 50-80% charge only, especially if it is left plugged in.

i never charged my phone during off state since i need it running all the time for notifications, etc. Batteries nowadays are far more reliable and robust compared before unless its a low quality/low cost like what happened to S4 last year, gets bloated.