Hello,
I think you seem to be talking about petechiae. They are not caused by drinking coffee. Purpura and petechiae are common after skin trauma and viral infections. They should go away in a few days. But if they persist then please consult a physician and get other causes ruled out which include allergic reaction, autoimmune diseases, thrombocytopenia, ITP or leukemias.
It is very difficult to precisely confirm a diagnosis without examination and investigations and the answer is based on the medical information provided. For exact diagnosis, you are requested to consult your doctor. I sincerely hope that helps. Take care and please do keep me posted on how you are doing.