T.:
My advice is going to be pretty blunt, but I think you need to impose what you think is best and not worry about what a 3-yr old wants or does not want. I don't think the sippy cup is the biggest issue, it's just that it's not good for him to get used to drinking all the time. It's a bad habit, which could lead to later problems with diabetes, overeating, etc. It simply needs to be stopped, there is no need at all for any child to be sipping liquid all the time. He needs to be taught to come to the kitchen to get a drink when he is thirsty. If the drink comes in a sippy cup, so be it (I do think you could slowly start thinking about introducing a "big boy cup"), but he simply needs to learn that going around sucking on objects is not acceptable.
If we as mothers fail to teach these important lessons to our children, how do we expect them to grow up and become self-regulating, mature individuals? you just have to say no, and stick by it. He may cry and kick and scream, but he needs to know you are serious. You are NOT going to damage your child at all by taking charge of his upbringing. Mothers have been doing that for thousands of yeas, they've just recently started worrying about the opinions of 3-year olds instead of their own decision-making abilities.
Hope this makes sense to you, please consider doing it. I have a boy (7) and girl (5) who are happy, well-behaved, a delight to be around. I've been a very strict and loving mom since day one, and I've never regretted making my children kick and scream a few times. The thing is, I seldom have to do it any more, since they know I mean what I say. They are lovely children, and I love to be around them, but they do know who's the boss (not them :)
Good luck, and stick to your guns. He'll love you best for that.
M.