Flies don't burp, nope! Which is reassuring to me, as it's common knowledge that when they land on your food they vomit in order to allow them to digest it, so I'm happy to know they're aren't burping too!
Burping (sometimes called a belch, gas, wind, ructus or eructation - though the latter probably not so much!) usually occurs during or shortly after eating. It's not a particularly complex process. When we eat food we also ingest gaseous air, and when we swallow these gases, they need to find a way out...
The ways that flies eat and digest food is different to humans- as I mentioned earlier, this starts with vomiting- and the make up of their bodies means they do not need to burp.
If they did burp you'd imagine that because they are so small, we wouldn't hear it... (unless they had been feasting on coca cola, maybe). I think the buzzing they make is annoying enough so I'm pretty glad about this!