I've killed a tomato ad squash before because it is hot and I got it too close to the plant. so now I only do that in the fall so it has time over winter to break down. I till it in so it breaks down pretty fast.
It depends a lot on your situation and what it is composed of. How often do you clean it out and how much volume are you talking about?
Some people, especially in the city, might bag it and ship it off to the landfill. Some people use it to make compost. I clean mine out once every three or...