Don't you eat? [Do you not eat?]
This is a negative interrogative. There is one verb in the sentence, to eat, conjugated in the second-person-singular present tense. To eat can be transitive (takes an object) or intransitive (does not take an object). Here it is used as an intransitive verb; the question is not asking if you eat some specific food, but only if you eat in general.
Don't you like to eat? [Do you not like to eat?]
Again, this is a negative interrogative. The main verb in this sentence is to like, again conjugated in the second-person-singular tense. But to like is always a transitive verb; you like some thing. In this sentence the question is whether you like to eat, which is an infinitive representing the action of eating. To eat serves as the object of the conjugated verb to like.
(The question also could be phrased "Don't you like eating?" which would use the gerund form of the verb to eat.)