Microwaves are a form of non-ionizing radiation, meaning it does not have sufficient energy to ionize an atom or molecule (i.e. remove an electron). Other forms of this type of 'radiation' are visible light, radio waves, uv light. Microwaves are capable of producing heat (hence the purpose of a microwave oven) however it cannot cause any sort of mutation that would cause cancer like x-rays, gamma-rays, etc. If the micorwave oven has been damaged so that the oven can be operated with the door open or if the door is cracked, then it is possible that you could experience slight burning in your fingers (which could lead to cracking etc). 

If you're worried about it my advice is to either a)buy a new microwave oven-they're pretty cheap nowadays or b)just wait a few seconds after the oven turns off before grabbing your food. I doubt that this is the case though, because when the oven *dings* off then there is no more radiation being admitted (it travels faster than any human reaction). Maybe your not letting your food cool sufficiently and burning your finger tips as you take it out.