The ESC controls what power the motor draws from the battery.

To burn an ESC your motor could go bad and burn the ESC, or other issues like a short in the wiring could burn the ESC but typically batteries unless they are crappy will not burn your ESC.

If you have a power hungry motor and large ESC with a less than adequate battery (opposite of what you asked) you could do damage, which you will notice by how warm the electrical components are.