Will playing a game in a Android phone while charging damage or cause harm to the battery or the phone?
Playing while charging does not damage the battery or phone directly.

The only problem is that:

battery gets warmer when is charging, and phone also gets hot when playing. So, when you play while charging, phone gets too hot and we know that battery life would be affected when it works in very hot conditions.

Sources to read more:

http://batteryuniversity.com/learn/article/how_heat_and_harsh_loading_reduces_battery_life

http://batteryuniversity.com/learn/article/how_to_prolong_lithium_based_batteries