Wasps keep caterpillar populations down, which would otherwise decimate our crops. If you're asking what animals could be safely subtracted from the food chain, besides humans, which even then would impact cockroaches, rats, pigeons, coyotes, seagulls, etc...
Apex predators could be taken out of the equation without too much of an impact. These are animals such as tigers, killer whales, bald eagles, polar bears...things which nothing preys on. The prey species would be overpopulated, but starvation or whatever condition killed off their predators would re-establish equilibrium. Like when wolves were eliminated from the US east coast, deer became a big problem. Apex predators are fragile species, and are also the ones people least want to see go extinct. The exception to this is sharks, which have survived many mass extinctions, even though there were long periods of earth's history where sharks weren't the top of the food chain.
EDIT: Pandas. There's a reason they're not fucking. I would say sloths provide food for slacker anacondas. And mosquitoes are very essential, its the only way complex proteins make their way down the food chain. DDT wasn't chemically affecting the eagle's eggshells, it was just that the fish the eagles ate weren't eating enough mosquitoes, so the eagles couldn't get all the proteins they needed to form strong eggshells.
Species I wish would go extinct? Hyenas, hunting dogs are much more efficient anyway.
Most necessary species? E. coli.