This is a good question but difficult to simply answer yes or no.
As the LED display technology goes, normally we do not need air conditioner system for the outdoor LED screen except the extreme installation environment. LED screen is very sensitive to the heat and working enviroment. For outdoor LED screen of permanent installation in tropical area or extreme hot climate areas like Middle east, with air conditioner, the screen can work in adequate temperature to avoid overheating and other relative problems to get longer lifespan and much less maintenance.
Below with a reference list to show how much cooling(BTU) we need for the LED screen base on the sizes.
There are other factors need to be considered if we go with air conditioner system.
1. The cost of adding air conditioning system
2. The cost of the structure of air conditioning system
2. The cost of more power consumption
4. The assembly and maintenance cost