I haven't tried this intentionally. I have accidentally hurt or killed grass with a good hay tarp laying on it for a prolonged time. My intention was to dry the rainwater off the hay tarp before folding it up to store it.
The biggest consideration is to adequately weigh the tarp down, especially around the perimeter. In my experience a large tarp can move around quite well in stronger winds. It may be a good idea to check on tarps during or after a particularly windy day. You may want the tarp to have a slight slope to avoid water pooling on it, especially if mosquitoes are a major concern.
My garden is composed of four major garden plots. My usual winter preparation consisted of covering them with organic matter, mostly leaves, 4-6”deep, mostly to keep the winter rains from beating up the soil. Recently I bought 5-mil “silage tarps for three of those beds. I don’t use them like JM does, I’m not a market gardener, but I’ve started covering some of the plots with the tarps to protect the soil over winter. Come spring when I’m ready to start planting, I remove the tarps and the beds only need a light harrow and raking and they’re ready to transplant into. So far I’m liking them. So it depends on what your trying to accomplish.