I don’t know why nobody quoted the sources instead of interpreting what they say. Here is a partial quote from Star-K on their website:
“Full-service Starbucks stores offer non-kosher breakfast items, like bacon and turkey sandwiches. While this is not a revelation, it may surprise you the extent to which this affects the kashrus of Starbucks coffees. These treif meats are served on ceramic dishes. The dirty dishes are washed in a three- compartment sink along with the brew baskets used to make the coffee and the small metal milk pitchers used to steam milk for lattes. It is this clean-up procedure that has soured many kashrus agencies on Starbucks coffee. While leniencies exist to allow the coffee in spite of the clean-up issues, Rabbi Moshe Heinemann, shlita, Rabbinic Administrator of the Star-K, has decided against using them.
But take heart, for all is not lost. Firstly, the above is true only at full-service Starbucks stores that serve treif sandwiches and the like. However, at the smaller Starbucks kiosks, typically located in airports and train stations, sandwiches are usually not offered. The good news is that the coffee making equipment at these establishments remains free of treif blios (non-kosher absorption), and you are permitted to drink the coffee there.