Is it safer to wrap or cover your food when warming them in the microwave?
Many plastics used in the kitchen are unsuitable for use in contact with hot food, whether wrapping it or sagging onto it when used as a cover. Freezer bags are a relevant example. These may melt, or leach into the food. I don't worry about this as much as I probably should; some people are very cautious. 

A sealed wrap is also a bad idea, as it can burst under steam pressure. For example leaving a bag clipped shut. This is rare though, and also applies to an unvented cover of cling film. 

A loose or vented cover is a good idea for keeping the microwave clean.

Your friend is misguided. Microwave ovens use a form of radiation similar to radio waves, and this is what heats the food. It's very different to ionising radiation which we are rightly cautious of. 

Microwave radiation isn't blocked by anything you'd use as a microwave cover/wrap, because then your food wouldn't heat up. So it's a good thing your friend's cover does nothing.  Microwave radiation is blocked by metal, including the mesh in the door. The emission of microwaves stops instantly when you open the door, because we're made of meat and would cook quite easily (plus the WiFi would be jammed). 

All links are from Wikipedia and are further reading for the curious.