You don't burb a sleeping baby. I never burb in the middle of the night after about 4.5 months. I read somewhere you don't really need to.
I used the baby whisperer method around the 5 month mark with my son. I got him to stop his waking. He started doing an 8 hour block. What I did was start with the first waking. I would just go to him and tell him it was time to sleep, not eat. When he was younger, I spent a lot of time patting his butt. So, I would pat his butt to sooth him. If that didn't work, then I would pick up him till calm then put him down. I did that drill over and over again and again, and eventually, after about 3 weeks, he was only waking the once. Around 8 months, he started waking for a second time, so next week I will be back to breaking him of that.
You aren't sleeping because your body has entered sleep deprivation hell. Once you get your son to sleep better, you will be able to sleep again after being woken, but right now, your body is in "why sleep?!?! I will just get woken up!"
Hang in there. The 6 month wonder week should really let him self-sooth and hopefully you can then get some sleep.
Sleep is a habit, so you just need to start teaching him to not wake. Many babies do need to be feed once or twice a night till 12 months, but they should be doing a good solid 6-7 hours by 6 months. If they aren't, you need to break the wake habit and teach the sleep one.