Case 1: In this case the food is taken from a particular location, so:
He eats from the plate.
Case 2: There are two options. Either you can describe the location of the food, or, again, from where the food is taken:
He only eats food (served) on a plate.
He only eats from a plate.
The second sentence is not unusual in context:
My cat is incredibly fussy. He will only eat food from his special china bowl. Otherwise he just turns up his nose and walks away with his tail in the air.
(Edit) Another way to say this is to eat off of a plate, which is synonymous with eating from a plate:
My cat will only eat off of china dishes.
Again, this describes where the food is taken from.
"Off of" does have other meanings, so context is important to avoid confusion. For example, let's say your town has a particular place where most of the restaurants are, we'll call it "Restaurant Row":
He likes to eat off of Restaurant Row.
This means he prefers restaurants that are not in this particular location.