I usually bury mine. Once I had a very small coop with bantams in it and a little house for them. A weasel got in and killed everything inside. I burnt that entire coop, including their remains. (It's kind of funny to me now that I just flat out burnt this thing that my family and I had made ourselves and spent a lot of time on, but at the time, I was very frustrated.)
If I find tiny bits of remains from a predator in the weeds, I'm likely to leave them there, unless large parts of the bird are still there. Then I'll bury them.
If I have chicken eggs that I was incubating that died mid way through incubation or during hatching, I'll either throw them in the weeds or in the river near my house.