What would happen to a food web if the organism in the center of it completely disappeared?
Did you see the movie "2012?" You know all of that world wide destruction and the end of life as we know it stuff? If you completely removed a piece of a food web, the above scenario wouldn't happen. But bad things would for sure happen.
The simplest possible answer would be to say that the entire food web would collapse, and all of the organisms in the web would die of starvation. Let me clarify. I am assuming (because the question didn't state) that the center organism is labeled "plants." If all plants cease to exist, the entire food web (and all life on earth) collapses. That's because the plant life supplies the initial energy into the food web system. No plants means no food for herbivores. No herbivores means no carnivores. Omnivores are also hosed. Without plants to supply the starting energy to the system, there is no flow of energy through the system. No energy in the system means no life. I guess "2012" would happen.
A less apocalyptic scenario occurs if the organism at the center is a specifically named plant. Use "grass" for an example. Anything that eats grass for its food supply will die. Anything that ate that organism would then die as well. And so on. The web would likely stay in tact, but several food chains (web threads) would be destroyed.
An even more hopeful scenario involves adaptation. Get rid of grass again. Many organisms will die as before, but many will adapt to the food shortage by eating different plants. The food web will fluctuate and likely change shape, but it will largely remain the same.