Domesticated animals would starve or adapt and eventually evolve. As would animals like bed bugs and other critters that depend on humans, and the animals that rely on those for food. I imagine that deer and squirrels would become overpopulated, and animals relying on their food sources would starve, but eventually regain numbers.
Pollution from power plants and whatnot that deteriorate without human upkeep would have numerous effects on wildlife, particularly marine life.
This also doesn't consider why humans will go extinct, as that may have an impact on other life if we die out from disease, extreme climate change, pollution, war, etc.
It's hard to say about animals but coral polyps will benefit greatly from the lack of human impact on our oceans.
Deer the most
Cows the least
Wild predatory mammals will benefit the most. Domestic animals, such as dogs and cats, will lose the most.
Perhaps the world will be taken over by insects!
Because we humans are big and clever enough to produce and utilize antibiotics and disinfectants, it is easy to convince ourselves that we have banished bacteria to the fringes of existence. Don't you believe it. Bacteria may not build cities or have interesting social lives, but they will be here when the Sun explodes. This is their planet, and we are on it only because they allow us to be.
― Bill Bryson, A Short History of Nearly Everything
Elephants and other endangered animals will be the lucky ones. Domestic animals may revert to their original source. Probably bacteria will rule the world for a long time.
All non-human life and nonlife, especially the bees. It'll be the utopia that only exists in our dreams.
I was thinking rats might not thrive as well if there were no humans. Deer and other animals that are currently flourishing because we've eliminated many of their predators might suffer as well.
Besides a few dozen species though, I imagine the other fauna would probably be better off. Perhaps the biggest benefactors would be large mammals that have not yet gone extinct. For whatever reasons, when homo sapiens began to spread around the globe, many large herbivores (woolly mammoths, giant sloths, mastodons, etc..) went extinct.