When in contact with the candle flame, the plastic cup will burn, or not, depending on the properties of the plastic and of the material that is inside it.
  In the present experiment it is maintained constant the properties of the plastic cup. However, the same cannot be said in relation to the content inside it. When inside the cup is air, it doesn't allow an appropriate dissipation of the heat produced by the candle flame, because the air has a high resistance to heat transfer. Thus, when the plastic cup contains only air, the temperature outside the cup reaches the plastic inflammation temperature and it begins to burn. The same happens for the sand because it doesn't dissipate the heat produced by the flame.
  In contrast, the plastic cup doesn't burns when it contains water because the water has a high calorific capacity that allows a good absorption of the heat produced by the candle flame. The water ensures that the temperature outside the cup is always lower than the plastic inflammation temperature. That because it uses the heat produced by the candle flame to increase its temperature slowly (high calorific capacity). Try it and know better the properties of water!