There are very few things that I won't put in the compost. I don't put anything to do with poison ivy, because I want it out of the yard completely. I don't put plant parts from the jimson weed that grows in the front yard - the seeds have a strong drug and I don't want the dogs getting into it. The pitbull one time I think ate a small sprouting plant in the back yard and was kind of loopy for a while.

Dog poop, weeds, food waste (suitably decomposed in the lidded bucket first), it all goes in the compost. If you want it to cook a bit faster then put lawn clippings on top. Anything that was alive can go into the compost.