J.M.
I remember burping going on for quite a while, but here's a trick that sometimes worked with both my kids when they fell asleep before they burped. Hold your son up on your shoulder in the regular "burping position." If he is asleep and patting him on the back isn't getting anything out of him, hold him tight to your chest and lean over across your lap, so that he's basically lying down on his back but with you holding him tight. He's still pressed up against your chest too. Stay that way for a few seconds, then bring him back upright and try burping him again. Sometimes we found simply changing the position they were in, even when asleep, could move the air bubbles around and they could get a burp up without waking. Worth a shot, right? Good luck!