You'd have to dry it completely -- which is surprisingly hard to do so it probably wouldn't happen -- to get it to burn properly. If it's even a little green/damp it would burn poorly, if at all, and smoke like crazy.
Assuming you don't let the grass get too long you can leave clippings on the lawn to provide a layer to trap moisture and eventually break down. But the key here is you can't let it get too long so that means mowing more often. I've had composting mowers and they tended to get bogged down in even moderately length grass. So I now just run my regular mower without a bag. There's a couple shaded spots that always grow really fast so either I put the bag on there or rake afterwards.
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