Just in case you didn't know, it takes 10-14 days for the dairy protein to clear both your systems so you can't expect much improvement before then if that is contributing to the problem. I did notice an improvement with my daughter's gas when I cut out liquid dairy (milk, I was drinking an insane amount of milk). I still enjoyed moderate amounts of ice cream, cheese, yogurt, etc..."processed" dairy. I started added milk back in around 4 months.
Breastfed babies MAY not burp or spit up as much, but to say absolutely that they don't is a little crazy. My midwife/lactation consultant said that, just like everything else, each and every baby is different. She is a breastfeeding mother to five boys and she said 4 of 5 spit up all the time.
You don't say how old your son is, but I encountered overproduction between 3 and 6 months. It was when she started spitting up a lot more often....like every time. The foremilk is watery, sugary, and gas-producing. I would suggest trying to nurse just one side at a session. Let him nurse until he breaks the latch, burp him, and then put him right back on that side. It will get more hindmilk in him and help settle his tummy. Offer the second breast if he is still hungry. Start with a different breast at each session.
Good luck. This could just be one of many bumps in the road.