Do other countries in the ME drink tea or coffee?

Yes, most people agree coffee actually originated from the middle east (Yemen). Tea is very popular in all of the middle east, with Turkey having the highest per capita consumption in the world at 2.7 kg (in 2010).

Edit: Here is some information about how tea reached the middle east

''Tea was most easily transporte d in the compressed “brick” form, and travel it must. According to the historical timeline in Microsoft‟s Encarta, “According to Chinese legend, the wearing of silk began around the 27th century BC. During the Later Han dynasty (AD 220), Chinese silks were carried by camel across the Gobi desert to the Tarim oases and on through Central Asia to Dimashq [Damascus]. The Chinese shipped porcelain and tea to the East [Ed note: east from a European perspective, meaning the Middle East], but in honor of their chi ef export, the route was known as the “Silk Road.” Chinese goods were traded in Dimashq for wool, gold, and silver. Some of the silk was shipped to Italy, where it decorated the togas of Roman aristocrats. Recent discoveries of silk strands in an ancient E gyptian tomb suggest that the silk trade may actually have begun as many as 2000 years earlier than scholars had previously assumed.”

A Short History of Tea by Paul Rogers