No, bottlenecks don't cause any damage to your PC whatsoever.
The main focus of the 'Bottleneck effect' is when a piece of Hardware, normally CPU or GPU are working with a huge performance gap between them.
Be the GPU working much faster than the CPU or the other way around.
As for a game to work properly, both CPU and GPU have to work "at the same time" so if your CPU or GPU is much slower than the other one, it will drag down the other until it finishes it's process to continue it's work. That's essencialy it.
Bottleneck just make your PC don't work at it's full capacity, because something is dragging it down. But no damage at all
Your FX-6300 isn't a great CPU and will probably bottleneck most recent GPUs, if you ever play an CPU-Intensive game ( It's a type of games that demands more from the CPU instead of the GPU. ). If you ever overclock the 6300, that way you can shorten the performance gap performance a lot, almost erasing any bottleneck.
A easy and nice way to see if there's any bottleneck happening is to download a System Monitor program, like HW-Monitor, it will monitor your system and show the Load on each piece. If ever the CPU or GPU reaches near 100% while the other is not as that high, that way you can say that the CPU is bottlenecking the GPU or the other way around.
Hope to have explained, I'm nowhere near a Hardware specialist but if you have any doubts feel free to ask.