Is it okay to leave a central air conditioner on all day?
The short answer is "no." A central air conditioner should only be left on when it's needed, and leaving your AC on all day can reduce its lifespan.
For most homes, the temperature in the house will not change significantly throughout the day. The setpoint of an air conditioner is usually between 74-76 degrees Fahrenheit, so if that temperature never varies, then there is no reason to leave your AC system running all day long.
This means that your AC system will run less, which reduces energy costs and increases your unit's life. Also, turning off your central AC system eliminates any chance of mold growth inside the ductwork.
A sized and installed central air conditioning unit will usually be able to efficiently cool your home when called upon, even in hot weather. During mild weather, you may not need to run the central AC unit at all during the day.
And finally, if your central air system is left on too long without cooling people and surfaces in the house, it can cause humidity levels inside the house to increase above comfortable levels... While only uncomfortable for humans by increasing relative humidity levels, this can quickly dehydrate wood or other materials over time. This means that leaving an AC unit on continuously will eventually rot out floorboards or damage furniture.