Risk of Fires
Metal and paper dishes and containers pose the greatest risk in the microwave. Microwaving metal causes it to spark, which in turn can cause a fire.
Look out for dishes that have a metal trim or edging as well. Some paper is treated with a chemical that can cause it to combust in a microwave, so avoid microwaving all paper as a general rule.
Also avoid microwaving wood as it is flammable.
Risks From Plastics
The FDA tests plastic dishes and containers to see if the chemicals in them seep out when heated. If it has the "Safe" stamp on it, the FDA has approved it because the seepage level is very low or non-existent.
If it does not have the stamp, that means that when the FDA tested it with heat, chemicals from the plastic seeped into the food, which is true of containers such as cream cheese tubs and yogurt cartons. None of these containers should be placed in the microwave.
Risks From Glazes
Most ceramic dishes are treated with a food-grade glaze and are perfectly safe for microwave use. However, some ceramics are treated with glazes that contain lead and other metals that pose a health risk.
In the US, ceramics that have a non-food-grade glaze are marked with a warning label so that they are not used for food. However, if you are unsure of the origin of your ceramic dish, do not use it for food or in the microwave.
Damage to Dishes
Non-microwave-safe plastic dishes may melt and become ruined if microwaved. The melted plastic can damage the inside of the microwave or burn you if your skin comes in contact with it.
Also, while most glass is safe to microwave, very thin glassware, such as wine glasses, can crack when heated.