Breastfed babies often don't have as much gas as bottle fed babies, they just don't feed as fast and take in as much air. I don't know which applies to you. That being said, placing him on his tummy is fine for having him burp. I was taught to sit my babies up on my lap and with my hand cupping them under the chin lean them slightly forward while patting their backs. The pressure on the tummy helps bring up the gas, and lying him on his tummy would likely have the same effect. Sounds like you are doing just fine, and if he's falling asleep after than he's getting the gas put and feeling comfortable.
I had trouble with my guy when he was born I would tickle up the back of his spine and then pat and just do that over and over again lol it seemed to work for him
you want him to burp because it causes gas and when he gets a lot of gas he will cry a lot with my son i try to rub his back in circles with your hand and start patting his back see if that helps him burp just keep repeating it rub in circles then pat.it sometimes takes a while but its worth him not getting gas.hope this helps