What is the proper use of "do" versus "are" in the phrases:
Do you like me?
Are you like me?
Is there any difference between them?
The misunderstanding is not about "do" versus "are".
Instead, it is caused by the word "like". It can be either verb or adjective, but also may serve an adverb:
Not very grammatical but a nice example sentence is:
Like my brother, I like fruits, like apples or bananas.
Here, "like" serves three different roles, and the meaning is:
Similarly to my brother, I enjoy fruits, for example, apples or bananas.
Now, back to the question.
"Do you like me?" means an asker wants to know if you enjoy the friendship of him/her.
"Are you like me?" means an asker wants to know if you are similar to him/her by character or whatever.
"Like" has two meanings.
When used with "do," it is a weaker form of "love. "Do you like (weaker than love) me?"
When used with "are," it means "similar to." Are you like me? Are you similar to me?