Farms deal with most of their animal deaths by engineering them. It's only when they're unplanned that they have to rent a backhoe.

As you guessed, the zoo has to pay to have the dead hauled off. They pick them apart, to see what made them tick (and to see what pathology was inside) and sometimes they save pieces (especially from endangered species) for educational purposes, but they incinerate the rest (which has to fit into smallish bags) Fun times out back with big handsaws.