What happens inside mushrooms when you reheat them?
Is it always unsafe to reheat them or is there something you can do to safely reheat them?
In France, white button mushrooms are called champignons. Before refrigeration was common, books suggested not to keep cooked mushrooms or reheat them, because undercooked mushrooms would quickly spoil. If you cooked them and later keep them in the refrigerator, it is safe to reheat mushrooms. Treat mushrooms as you would treat meat.
Mushrooms are mainly water, so they reheat well in the microwave. Do it on low power. Depending on the cooking technique, the texture may change by using high power.
It depends very much on which mushrooms. For button mushrooms, see papin's answer; this does not however hold for wild mushrooms.
In particular, some wild mushrooms (in particular those called "le champignon bleu" in French, I never found out what they were in English) may cause serious poisoning if eaten 24 hours after cooking. Believe me, I know. Apparently they contain a protein which changes after cooking, and one day is enough for it to become poisonous.