Did the Ottoman Empire ever ban the consumption of coffee with milk? I know Sultan Murad IV banned drinking coffee by itself, but was there any laws on drinking coffee with milk? If so, what was the punishment for it?
You may be mixing Murad IV's short-lived ban on coffee with a general distaste for coffee and milk in many late medieval/early modern cultures. The origins of this distaste for coffee and milk may have stemmed from Islamic alchemy. Taking from Stewart Lee Allen's The Devil's Cup:
The second part of [an ancient Ethiopian] ceremony, where the roasted beans are added to milk and imbibed, indicates it predates Islam (A.D. 600) because Islamic alchemists believed that mixing coffee and milk caused leprosy (a belief that lies at the root of the disdain many Europeans have for coffee with milk).
Adding milk to coffee seems to have become popular among Europeans in the 17th century:
According to Diglas, Vienna is also where adding milk or cream in coffee first became common. This, however, is conjecture. All we know is that it was a European innovation, because the Turks (like the Hindus) believed that combining milk with coffee caused leprosy. We also know that early London coffee society did not generally use milk. This leaves the Italians or the Viennese as the most likely innovators, since both were among the earliest coffee drinkers in continental Europe. Diglas pointed out that the two countries have milk-based brews, both completely different but bearing similar names— cappuccino from Italy, and Vienna’s kapuziner.
Allen notes that coffee was banned in Mecca in 1511 and 1525, Cairo in 1539, and of course in the Ottoman Empire in 1633 by Murad IV. However, the motivation for these bans is a combination of social control (as Yannis Rizos pointed out) and Islamic restrictions on intoxicating beverages. Allen discusses the 1511 and 1633 bans in some detail, and milk does not come up in anyone's reasoning. It therefore seems unlikely that the Ottomans ever specifically banned drinking coffee with milk.
Note: The internet is filled with references to Islamic alchemists believing that milk and coffee cause leprosy. Some pages cite Allen, some cite no one. Unfortunately, Allen doesn't cite a source for this claim either, and in fact only mentions it in the two asides quoted above. I'd really like to see some Islamic sources supporting this claim.