They do mention guns in Mass Effect do run out of actual ammunition and need replaced after some time. The standard M-8 Avenger ammuntion block supposedly lasted around 50,000 rounds. As for heat sinks, they were essentially just cylinders made of heat conducting materials that the heat was directed torwards. Ejecting one and putting a new one in simply allowed the gun to continue to fire without damage. A gun could continue to fire if not using one or one that was already used, but the heat would quickly start to melt the barrel and other components. As far as Destiny is concerned, handcannons (and some others) are likely energy based weapons. The Thorn is a prime example. Its rounds are as large as the gun itself.