Does Arduino Uno R3 require cooling in a closure?
The only times I had an AVR run hot was when I applied reverse power to it. You'll be fine without a fan for the Arduino board itself, as long as you don't run power hungry peripherals from it.

Probably the only thing you really need to check (apart from absolute maximum ratings: 40mA max per IO pin and 200mA max for the whole microcontroller) is the on board voltage regulator (usually three pin device with a heat sink soldered down to the PCB) if you power it from a voltage > 7V or so.