I think it depends on the child and on what you mean by "funeral". When I was a kid my folks took the animals to be put down and it was hard for me to "never" see them again, I didn't have closure, didn't see the full cycle of life, a "funeral" would have helped me get over their loss sooner I think.
With my kids we had a "funeral", or rather a ceremony, (even if the vet did the deed) where the kids dug the hole, buried the animal, thanked it for being a good pet, covered it up and made a little marker. They seemed to need the ritual to help them let go of the pet, not mope, and move on to a new animal.
Do what is best for your kids, just don't let them make a big deal of it regardless of what you decide to do.