No chance whatsoever. Not all radiation is the same kind. The light you're using to be able to read this is radiation, but you didn't ask about that. Why didn't you?
The trouble with the word "radiation" is it covers a whole number of things that can travel through space. Which is not helpful for understanding what's going on. What KIND of radiation? Some is dangerous, most isn't.
What is definitely dangerous is ionising radiation, basically radioactivity, gamma rays and X-rays. These have high enough energy to damage the atoms in your body by knocking bits out of them and causing cancer. Which is why you don't really want to have an X-ray photo taken in hospital too often, and why the radiographer hides behind a screen while they're doing it. I had one done a couple of months ago. I could hardly breathe because something was clogging up my lungs so he wanted a chest X-ray. One quick flash like that is OK, but the radiographer is doing dozens of them every day and needs to stay safe. (It turned out to be just an infection and a good dose of salbutamol - people with asthma get that as inhalers - plus a week of antibiotics put me right again.)
Microwaves, on the other hand, are a specific frequency of lower-frequency electromagnetic radiation (like light and radio waves). They are at the right frequency to make water molecules jiggle about, rub together and get hot, that makes the rest of the food get hot too, bingo, hot food.
The only possible danger from a microwave oven is if the door does not shut securely, so microwaves leak out and cook YOU. What prevents this normally is a metal grid on all sides of the oven, including in the door window, This creates a Faraday cage and the microwaves can't get through that, so you're safe.
And never put anything metal in a microwave oven. The microwaves cannot penetrate through it, and you can get sparking from a build-up of electrical charge. In extreme cases you will get a spark of lightning back into the cavity magnetron (which generates the microwaves) and now you have one very dead oven. If you've ever wondered why microwave ovens are very heavy at one end, that's where the cavity magnetron is - it contains a heavy magnet. The rest of it is basically a Faraday-caged box to put the food in.
My physics degree.