Plants existed before animals ever evolved, and if all animals were to disappear, plants would continue to exist a million years from now. Just mostly different species.
Plants produce both CO2 and O2. Without animals, there would be a higher concentration of O2. If you wanted, you could easily evolve plants that have internal processes that take in environmental oxygen at a higher rate than they use CO2, and perhaps use it to increase mobility and growth. Otherwise, it's likely that bacteria would take up any slack, assuming you just didn't re-evolve animals (Oxygen is a great energy source, if it doesn't kill you first).
Insects and other animals are only one mechanism by which plants are pollinated. Wind dispersal is an older method, and one that most pine trees use effectively. The plants that cause most spring allergies rely on wind distribution of their pollen.
In a compost pile, worms are famous for doing the work of breaking waste vegetation into new soil. But this happens even without animals. Instead, soil bacteria and fungus do the job. I wouldn't even suggest that the process needs to slow down.
There are lots of plants that rely on animals carrying their seeds, either as food (eaten or stored) or by burrs attached to fur, feathers, or skin. But there are many, many plants that rely on other strategies. Fluffy feathery seeds floating in the wind are common and maple trees with their helicopter seeds are two examples. Seeds in flood areas can use flooding both as a means of being carried away from their parent, and as a signal that it's time to germinate. More unusually, there are even plants whose seed pods explode (video), propelling the seeds many feet away.
Competition and predation.
With or without animals, plants need to defend against predation and competition. Dodder and mistletoe are both parasitic plants that get their energy by tapping into the sap of host plants. They don't usually kill the host, but they can certainly weaken it. Strangler figs are a species that germinate on a host tree, then as they grow, they wrap around and choke the host to death. Oak leaves (and others) contain tannins that poison the soil at the tree base, making it harder for other plants to grow there. It's a violent world out there.
If you also eliminated fungi, many, many plants would struggle because they rely on fungus at their roots to increase their nutrient uptake. Without that, plants would probably be limited in size, and many existing species would die.