They're both correct, as the others have mentioned, but "Do you like to read?" sounds more natural to me.

You can use both :)

both are correct but they are used differently. Do you like reading? this is used to refer to a general habit or something you always prefer to do i like tea.( i always do as tea is my favourite drink) i like reading. i would like to read. ( now) as i'm bored of work so i want to read something now.

'Like' is one of the few verbs that can take either the -ing form (gerund) or the infinitive as a verbal direct object. So both forms are correct, although we tend to use the -ing form more when we are talking about an activity in general.

Do you like to read ? is a direct question, do you like reading is a begining of a question "Example" do you like reading history or novels or do you (like) to read at all?! kind of asking do you ever pick up a book.