Addicted : This is a big problem. Kashrus wise. I should know. AS a girl I worked at a nonkosher coffeeshop. (before my days of making teshuva).
In these establishments they also sell sandwiches. One popular one is ham and cheese. When the plates or knives, or anything- needs washing, they put it in the same dishwasher as the glass heat proof coffee pots, which catch the perculating coffee!
This is a BIG Misconception, that we can buy black coffee in a disposible cup. People need to think Whats going on before the coffee hits your disposible cup? The stainless steel coffee filter that holds the grinds is in fact washed out in a sink/dishwasher that serves up pork as well.