My son managed to damage the charging cable. He pushed something against the charger, the cable was bent back and something shorted inside.
It was charging his ipad at the time and damaged the charge circuit so that it would only charge when switched off. When up and running, it would show it was charging but in reality wasn't.
So, from that lesson I would say yes. If you have a problem with the charger and it shorts inside because it's badly manufactured, then it can definitely damage your phone.
And naturally, it could electrocute someone or set fire to your house.
These things are often left plugged in for very long periods of time and forgotten about, so plenty of time to overheat.
I wouldn't go so far as to say only buy Apple, but there are reputable and cheaper brands out there (Ravpower, Belkin, Ankar... for example) and only buy MFI certified. (They last longer too).