"I do like" is mostly used to contrast with an earlier "negative" statement. "I don't like .... . I do like ..." This is true for most cases of "I do [infinitive]". It could also be to contrast with what someone else said: "I don't like A." "Oh, I do like A." So it's all about contrast. "do" will be stressed when pronouncing sentences where it appears.

Tony: Is that a new hat? Mona: Yes, it is. What is your opinion of it? Tony: I LIKE it. Mona: Are you telling me the truth, or are you just saying that because you do not want to hurt my feelings! Tony: Believe me. I DO like your hat!

In addition to the meanings given below by Meryn and James, one also can use 'do like' as an emphatic way of saying 'I like'. So, for example, someone could say, "Oh I do like your hat!", meaning, "I really like your hat!".