Bottleneck is a terrible word. CPU and GPU performances are completely different.
CPU can not limit GPU performance, though it can limit overall performance in very some number of cases.
We are talking about running a Core 2 Duo 8200 and a 280x or something of that nature. Then the CPU will be working at 100% in a new game like say Dragon Age Inquisition and limit the overall performance.
CPU's and GPU's that are small generational difference and similar range can never really bottleneck each other.
The I3 540 won't bottleneck a 270 card. We would be talking bottleneck at around 290 and better card.
I think an old i3 like that will be a bottleneck in a significant number of games, but so will a R7 270. It just depends on the game, image quality settings, and resolution. The higher the resolution and image quality settings, the more the bottleneck will shift away from the cpu towards the gpu.
But to answer the question, no it will not damage the computer. The only way it could is if one of the components overheated, and that should not happen since either the cpu or gpu should throttle if it reaches dangerous temps.