At work, I occasionally heat up coffee in a non-kosher microwave (used heavily by non-Jewish co-workers), by covering the top with a paper towel. Today, after microwaving my coffee, I noticed the paper towel fell off inside the microwave. Was the coffee ok to drink? Is the mug still kosher? Is it ok in the future to simply cover a mug with a paper towel when microwaving?
You should not use the microwave in this way.
The problem is that if used heavily, the microwave is likely to be dirty with non-kosher food, which will come into contact with the mug.
In addition, the paper towel is not hermetic, and the steam coming off from the coffee will also be a problem.
I don’t know what the state of the microwave was when you used it, and therefore cannot tell you the status of the mug; however, I would advise using a different mug (assuming that the loss is not substantial), and that you don’t make use of the common microwave.
For more details about microwave use (and in particular about using a non-kosher microwave) please see here.