Dogs don't need to "keep kosher." However, there are kashrut considerations when feeding your dog.
For example, it's forbidden to own or derive benefit from chametz during Passover. Therefore, if a dog food contains chametz, it's forbidden to own it or give it to your dog during Passover.
Another consideration: It is forbidden to derive benefit from milk and meat that were cooked together. Therefore, if a dog food contains such a mixture, it's forbidden to give it to your dog.
There are brands of dog food certified as Kosher for Passover, not because they are kosher for humans, but because they contain no chametz!