Although coffee generally becomes staler with time, heating coffee up in the microwave does not produce any harmful chemicals - as long as the container you are microwaving it in is microwave-safe.
The microwaves emitted inside the microwave are a safe form of radiation that cause the water molecules in the coffee to move faster, thus creating heat. This is harmless.
As far as taste goes, you may consider just making a new coffee, as this will always taste better, in my opinion!