J.,
I don't know that burping is necessarily the problem with him waking, but I think some children need to burp more than others (mine never really needed to), so I won't say that it couldn't be part of the issue. I wonder if another reason he is waking is because he is realizing that he's not nuzzled against his mommy and then doesn't know how to soothe himself back to sleep. He is old enough now that it would be a good thing for him to start to learn, or you will be setting yourself up for more difficult sleep issues than this!
I wouldn't nurse him to sleep at all. In fact, I'd do my darndest to keep him awake while he nursed - play with his little hands and feet, talk to him, etc. Then burp him or change him or do whatever you think will help him be comfortable and ready for bed. Hold him or rock him or sing to him or whatever until he is drowsy, but not asleep. Then lay him in his bed, give him a kiss and leave the room. If he fusses, go back in to check on him in 3 min, rub his head, put his pacifier in or help him find his fingers, or other things that you know he finds comforting, then leave the room. If he fusses again, wait until 5 min to go into him, comfort him again but do not pick him up, then leave again after a minute or to. If more fussing, wait 7 min, 9 min, and so on until he goes to sleep. Be consistent with this method for a couple of nights and he will learn to soothe himself to sleep, as well as get himself back to sleep if he wakes later in the night.
I did this with both of my children (my youngest is 7 months now), and they both slept well through the night at an early age. The longer you wait to teach him how to self-soothe, the more difficult the adjustment will be for him when you do. Honestly, 4 months is not too young. Good luck!