For a number of reasons we would prefer not to open our 20X40 inground pool this season. Can the pool be left closed without it turning into a swamp?

Is it covered and full?

Yes - mesh cover, water almost to top of pool (we're in NH - lots of snow melt!)

You will need to gets some chlorine in the water or it will get swampish and smell. Depending on your skimmer system you can periodically run your pump to recirculate the water.