Question: The word "dormitory" means "sleeping place". Is there any similar word for "eating place"?

Answer: 
There's also *refectory*: 

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> A refectory (also frater, frater house, fratery) is a dining room, especially in monasteries, boarding schools and academic institutions. One of the places it is most often used today is in graduate seminaries. It is derived from the Latin reficere: to remake or restore, via Late Latin refectorium, which means a place one goes to be restored. [Wikipedia]
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the root of which it shares with *restaurant*: 

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> Modern restaurants ... emerged only in 18th-century Europe, although similar establishments had also developed in China. A restaurant owner is called a restaurateur; both words derive from the French verb *restaurer*, meaning "to restore". [Wikipedia] 
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Oh, and of course *beanery*, *hash house*, *greasy spoon*, & *automat*.
There are lots of words for "eating place": dining room, cafeteria, restaurant, mess hall (military), galley (Navy), and so on.

Be aware that "dormitory" has a specific meaning, and is usually associated with schools, especially colleges. There are other words for sleeping place, including bedroom. If you're wondering if there are any words for eating place that are derived from the Latin meaning "eat" (as *dormitory* is from the Latin word for "sleep"), I'm not aware of any.