|MadSci Network: Environment|
That's a complicated question to answer in a few lines. As you may know, plants (not animals) are the living things that have the largest impact on the carbon cycle, by storing carbon through photosynthesis. However, ecosystems involve complex interactions between plants, herbivores, and predators and a change in any one group will affect the others. For example, removing herbivores can lead to more plants, while removing predators can lead to more herbivores which can lead to fewer plants. At first glance, removing all animals would lead to more plants and therefore more storage of carbon in plant matter and less CO2 in the atmosphere.
However, plants and animals don't just interact in terms of what eats what. Plants rely on animals to pollinate their flowers (bees, bats, birds) and disperse their seeds (birds, rodents, ants). Likewise, many small animals in the soil help to break down leaf litter, speeding decomposition and returning important nutrients to growing plants. Ecosystems, ultimately don't function without animals, and removing them could lead to the collapse of many ecosystems. With such a collapse, you might expect the CO2 to increase (because there could be a shortage of new individuals or nutrients to help take up the CO2). There's not a single answer to your question but it is an interesting exercise to think of each organism and how it is connected to all the other organisms in an ecosystem.
And who would want to live in a world without animals, anyway?
Try the links in the MadSci Library for more information on Environment .