Can you keep a 15x30 in ground pool shut down for the summer in Florida's hot weather?

I doubt it. Not without someone adding sanitizer (chlorine) on a regular basis. An alternative would be to add a Saline System, and leave the equipment operate for a few hours per day on the Timer. The saline system will produce all the sanitizer your pool needs, automatically, from a mild level of salt dissolved in the pool water. You would need an automatic topup (fill) device as well to counter evaporation.