He is doing what comes natural. That's like saying I have a teenager that is rolling her eyes at me. Yep, that's what they do. And it will be worse before its better, much worse at 3. Find a parenting strategy and stick with it. Love and Logic is pretty good. Read the book in your few mins of down time.
Keep calm and try to head it off before you sit down to nurse. Put out a snack, every time, before you nurse. Don't be driven by his whining. He says I'm hungry, point to the snack. I don't want that, point to the snack. Don't ever reward whining. Tell him the words to say. Tell him I can't understand you when you whine. He is very young so try something that can keep him by you while you nurse. Read a book with him, give him a toy he can only play with when you are nursing, a stuffed lovey especially for that time.
It's just a kid. You will either teach him TO whine, or teach him NOT to whine, by your actions. For years. This is one of the tough parts of parenting.