It really depends on the organism web. First off, we must know what we cannot eliminate:
1) Producers: Producers are the key to the whole food web, without producers, the whole web would collapse.
For the next, you can choose one or the other:
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2) Omnivores: Omnivores can technically be removed, for they can be replaced by herbivores and carnivores (Omnivores doing both of the jobs). They can also remain if we take out the other two, if we factor in humankind (since humans are supposed to be omnivores)
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2) Herbivores: Herbivores generally do a better job in making sure that vegetation does not overgrow, as they only eat greenery. They are also limited by carnivores (& omnivores if they are smaller animals).
3) Carnivores: Carnivores only eat meat, which would allow them to control the population growth of the Herbivores. As long as the Carnivores do not over-hunt, the food web would continue to be sustained.
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Finally, we arrive to the last part of the food web, the decomposers. If the decomposers are taken out of the web, the food web would indefinitely collapse soon, as dead organisms and plants continue to lay around, which could become breeding grounds for bacteria and diseases, which would destroy the whole web. Decomposers also return nutrients to the ground, which is vital to the cycle of life.
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So it really depends on the food web that you look at. Generally you would first eliminate the omnivores, and any extra carnivores/herbivores.
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