If you play too many graphically intense games on your PC will that burn up, slow down, or ruin it in any way?
No. Running games is just like running any other cpu/gpu intensive application. Games probably stress the processors more than most other application, but they themselves will not "ruin" a PC. I maintain a few gaming PCs in my house and they see a ton of use from myself and my kids. I have never had any problems with these PCs.
Would you say an i3 processor be at least ok?