Lightweight loadouts are king. I went airsofting on Sunday and had no issues whatsoever, but I was running a very lightweight loadout. Trousers, t-shirt, belt kit, no rig as I was using my VSR but could've run a lightweight chest rig if I needed to, and a simple cobra hood instead of a full ghillie. Keeping at least your back uncovered by extra layers, backpacks or load-bearing kit really helps you to keep cool as it allows you to sweat, then that sweat to evaporate which is how your body cools itself. If you have too many layers, you sweat but then that sweat gets trapped in those layers and you don't cool off as effectively (don't cite me on this, I am no scientist but this is what I've found through experience). I wouldn't want to play CQB unless there was some sort of environmental control to keep the temperature down and to cycle the air as it's going to get really, really stuffy in there really, really quickly; woodland is far better when it's hot because the air doesn't get stale and stuffy like it would at an indoor venue, due to being outdoors.
The most important thing, so important that I give it its own paragraph, is keeping hydrated. Your body uses water to keep your body temperature at healthy levels, so while dehydration is really bad, it's worse in the heat and can lead to heatstroke (which can be life-threatening faster than dehydration!). I think I went through about 6 or 7 litres of water on Sunday.
Fortunately, it typically it doesn't get too bad in this country. When I did airsoft in the south of Spain, we'd have to finish by 2pm because it just starts getting dangerously hot (hottest day when I lived in Spain I think was about 45 degrees celcius)