The image isn't my exact coffee cup, but it is similar. I didn't read the label and microwaved it until I smelled something strange (probably a minute and a half). Luckily, I when I took it out, only a small part of the plastic on the rim of the mouth was melted, but is it still safe to use for future use? I didn't see any damage done inside the mug.
Normally you don't put metal object in a microwave because it can make sparks and ruin the object, the microwave, and be potentially hazardous (though pretty nice, check https://www.youtube.com/watch?v=u_yg5eKjA4U for example).
Plastic doesn't absorb as much heat when metal does (induction phenomenom), so local heat started melting the plastic. As with any melted plastic, it's not necessarily good, but once cooled and washed I don't remember having thrown away melted bowls or plates, nor do I think it was specially advised in the past.
About this dioxin polemic, it's rather the fact that you shouldn't heat something directly in a plastic that may release dioxin when heated, which is not your case since I think your cup has had time to cool down since yesterday.