If you put trash in the recycling can, they might not dump it at all. We put some wooden fruit boxes (when you buy a case of fruit, it doesn't always come in a cardboard box) in the recycling can along with our newspapers and plastic, and the garbage collectors left us a note that they don't recycle wood.
Scarp wood can't go in the yard waste can, either, supposedly.
Bust the crate up and put it in a green trash bag. It will go to the dump.