I remember in third grade, we were often given an exercise to look up words in the dictionary - practicing alphabetizing. I was always slowest... because I would get distracted by reading the dictionary.
If school is so boring or unchallenging that the kid reads to escape, then they are wasting the kid's time when he could be learning something. I'm not sure it's a great idea to make the kid experience school as a boring frustrating place where he's not allowed to learn.
If it were my kid, I'd make a deal: First, pay enough attention to learn everything in class, as shown by grades on quizzes (e.g. over 90% or even 95%) and ability to answer the teacher's questions when asked. Second, class is for learning, and any book read in class should be useful. (My wife, who's a teacher, says for a 3rd grader any book at their reading level is useful!)
For homework, have conversations about work habits and whether it's better to interrupt work with fun or get it over with. Again, if the work is sheer repetitive drudgery with no intellectual reward whatsoever, then maybe it's worth escaping.
In fourth grade, we had self-paced workbooks for all the subjects. We were intended to take about a month per workbook. I found spelling so easy that I was doing several per week, and not spending enough time on other subjects. They pointed out the problem and made a deal with me that I should hold it down to one spelling workbook per week. I saw their point, thought this was quite reasonable, and was happy to comply. If they had tried to limit me to one a month I would have been very frustrated.
Richard Feynman's high school physics teacher, seeing he was not challenged by the class, gave him a calculus book to read in class and said he wasn't allowed to speak up in class till he had understood it. Writing about it later, Feynman was glad that this had happened.
Bottom line: Figure out a plan that respects your goals and his goals (if you want to accommodate the teacher's goals, at least one of you should actually value those goals). Discuss the plan with him. Make sure your plan accommodates his desire to learn and achieve. You will probably find that clear discussion of goals and purposes will lead to an agreement he will be happy to follow - and you may find that that agreement includes some reading in class.