Does a 3rd-Party Cable cause any harm when charging your Phone with the original Power Brick?
Typical connector theory applies.
In order for a smartphone cable to function it needs the following properties:


Proper connector within spec, otherwise it has bad contact or will damage the socket.  
Enough copper wire to handle the current. Otherwise voltage drop will cause charging to abort. This is especially a problem since USB cables were not intended to do >500 mA, until smartphones came around.
In case of iPhones: A valid ID chip. Otherwise the iPhone will say: "Accessory incompatible".  


Damage will only occur when there is a physical problem with the cable, or a polarity problem of course, but those are rare.

Most damage will happen when the cable or socket has wear. After many mating cycles, cheap cable contacts wear out causing higher resistance, with more heat. Sometimes the phone melt and causes fire.
This is because the springiness of the contacts degrades over time, and degrades faster when heated.  When you often use a slightly off-spec cable, it might cause damage in the long run.

Don't be stingy on a 10$ cable for your hundreds of dollars device.