why does the  paper cup containing water not catch fire on heating ?

This is because when we heat the paper cup containing water, then the heat supplied to the paper cup is transferred to water inside it by conduction. Due to continuous transfer of heat from Paper Cup to water , the paper cup does not get heated to its ignition temperature. Hence the paper cup does not catch fire.

The reason is that, perhaps surprisingly, the paper simply does not get hot enough. Paper has an autoignition temperature (the temperature at which it will burst into flame) of about 210-250o C. If you were to just put an empty paper cup on the hot steel, it would rapidly reach this temperature and start burning. However, with water in the cup the situation is different. The presence of water on the other side of the wall of paper will reduce the rate at which the paper can be heated by the steel because it will cool the material. This cooling can be fairly efficient since convection will cause hot water to move away from the walls, letting cool water replace it. Once you reach 100o C, the water will start boiling. However the temperature of the water will not rise any further at that point, because as happens at any phase transition, all the extra heat added to the system will go into that driving that transition (boiling). So surprising as it may sound, you can get into a steady state regime, where even if the water is hot enough to boil, because this process occurs at 100o C, the boiling water may still be "cool" enough to prevent the paper from going above 210o C(ish) and burn.

Paper burns at temperature higher then the boiling point of water. When water is heated in a paper cup, water circulates heat away from paper until it starts to boil at 100C. Once water starts boiling, it can’t get any hotter then 100C, so consequently paper can’t become any hotter then 100C. So paper cup does not burn because it simply doesn’t get too hot to burn.

this is taken from quora
 

because the heat will get transferred to the water by the process of conduction ..and due to the presence of water papers ignition temperature isnt reached.. 
 

the temprature of the paper cup does not reach its ignition temprature. Thus the paper cup does not burn. The paper cup containing water does not catch fire because water does not let the paper reach its ignition temperature, i.e. the lowest temperature at which a substance can catch fire.

This is because the water takes away the heat from the cup and does not allow it to reach its ignition temperature

paper cup do not catch fire at burning when it has water in it as the heat gets transfered to the water through conduction and as the ignition temperature of paper cup is not reached it will not burn. It will only boil the water in the cup