growth plates are the plates at the end of all of your long bones. they control how your bones grow during your adolescent years. the majority of growth plate injuries are fractures that are a result of an accident. the remainder of injuries (1/3) are caused by constant repetitious motions, such as a pitcher perfecting his pitch for hours at a time.
I really don't think you'd need to worry unless you plan on doing push ups for 2 hours straight everyday, but then you'd probably have bigger problems on your hands then your growth plates.
in short, my answer would be no. I doubt push ups will do anything but good. just remember to not over do it, and if something starts hurting stop. listen to your body, it will tell you when its in trouble.