Can I run my Arduino Uno R3 for 24/7 ? Will it be damaged by doing so? If yes where can i get a heat sink to cool down my arduino and how to do it? Thanks in advance.
Um. You arduino should not be "hot" enough to worry about a heat-sink.
Probably DC on-board regulator is hot, isn't it? If yes, then input voltage is high or the load/current is high. Both can be solved by decreasing of input voltage to about 9V. In case of high current it is more difficult. Arduino itself has low consumption itself but if you connect something like shields with higher consumption, displays, LEDs, etc., each one is increasing the consumption. Consider separate power for it. Reasonable current for DC regulator like UNO or Mega uses is up to 200-250mA (some boards use switching regulator which is more powerful). If even though you will need a heat-sink, try to look small adhesive heat-sink on ebay.
Generally, the ATmegas, like UNO R3 has, are feasible to run 24x7. Arduino UNO has 2 ATmegas and some other parts, nothing which could prevent to run the board non-stop. For an example, I am using not exactly Arduino but self-made device based on Arduino for heating system in my house. It will be 3 years soon it operates nearly continuously. I do not consider several blackouts.
"Will it be damaged by doing so?" Yes, sure. The Atmega is sensitive electronic device which can be destroyed easily under bad conditions. However, I remember older and more sensitive ICs, so from my view the ATmega is pretty durable. The memories are liable to wearing. The has limited number of write/remove cycles: flash 10k, EEPROM 100k. In simple words, it would be able 10k times write a new sketch. And about the EEPROM, it should not be used frequently. 100k writes to occasionally save some setup parameters is pretty much.