What animal or insect going extinct would have the greatest impact on the ecosystem?

Humans.

Concur, they didn't specify a need for the effect to be negative!
- It is what I meant…… but I’m very happy to see some folks thinking about it from a different angle haha.
ctrl-F humans

boosted and updoots to the right
That's really the only right answer. If we're looking at the impact of a single species, as opposed to a genus, family, or order (like most of the other answers are doing, e.g., "spiders"), humans are the only single species whose absence would cause vast changes in the biosphere. We have no other close taxonomic relatives that could step into our "niche" and continue doing what we're doing. Losing one species of mosquito (instead of the whole genus) or one species of plankton (instead of the entire.. god, what, order? Clade?) wouldn't produce any significant effect by itself.

The real answer is Humans

That book The World Without Us makes a great case for that.
The best thing for the continued survival of the planet and all other species would be the extinction of the human race. Sad but true.
- Extinction of capitalism. Humans existed/exist for thousands of years in harmony with their environment.
- Most other currently extant species yes, planet and life in general, nah. There's too many extremophiles out there that prove that even if we make the planet completely uninhabitable by anything even remotely resembling humans, animals, plants, etc, there will still be life in one form or another. Try and imagine what we'd have to do to screw up the planet bad enough that tardigrades would be unable to survive.

Oceanic plankton produces like half of the world's oxygen. Trees get too much credit. I'm not sure what the exact impact of losing so much oxygen would be, but... Not good?

Plankton isn't an animal or insect though, it's algae and bacteria

My vote goes to worms. Without them huge amounts (like the vast majority) of land will become dead after a few years. Worms are very underrated
- That’s an interesting take I don’t think anybody else has said worms yet
You and I thrive in oxygen, because we evolved in its presence, but oxygen is a really potent corrosive chemical that destroys a lot of life. When blue-green algae first showed up and started dumping oxygen everywhere, it in turn was a cataclysmic event for life on Earth.

https://en.wikipedia.org/wiki/Great_Oxidation_Event

The Great Oxidation Event (GOE), also called the Great Oxygenation Event, the Oxygen Catastrophe, the Oxygen Revolution, the Oxygen Crisis, or the Oxygen Holocaust,[2] was a time interval during the Early Earth's Paleoproterozoic era when the Earth's atmosphere and the shallow ocean first experienced a rise in the concentration of oxygen.[3] This began approximately 2.460–2.426 Ga (billion years) ago, during the Siderian period, and ended approximately 2.060 Ga, during the Rhyacian.[4]

The sudden injection of highly reactive free oxygen, which is toxic to the then-mostly anaerobic biosphere, may have caused the extinction of many existing organisms on Earth — then mostly archaeal colonies that used retinal to utilize green-spectrum light energy and power a form of anoxygenic photosynthesis (see Purple Earth hypothesis). Although the event is inferred to have constituted a mass extinction,[7] due in part to the great difficulty in surveying microscopic organisms' abundances, and in part to the extreme age of fossil remains from that time, the Great Oxidation Event is typically not counted among conventional lists of "great extinctions", which are implicitly limited to the Phanerozoic eon. In any case, Isotope geochemistry data from sulfate minerals have been interpreted to indicate a decrease in the size of the biosphere of >80% associated with changes in nutrient supplies at the end of the GOE.[8]

Probably be pretty bad for us, but I suppose if you're an obligate anaerobic organism, you'd be having the best situation since a couple of billion years ago.

Lets hope it's not rats!

HAHA.

I know it's already on the post but I'll say it.

Bees

To put it simply

"the availability and diversity of fresh produce would decline substantially, and human nutrition would likely suffer. Crops that would not be cost-effective to hand- or robot-pollinate would likely be lost or persist only with the dedication of human hobbyists."

"In Europe alone, 84% of the 264 crop species and 4,000 plant varieties exist thanks to pollination by bees. Some attribute the following quote to Albert Einstein: "If the bee disappeared off the face of the Earth, man would only have four years left to live.""

Bees are very important, you kill them off and Humans would die

We would be fine without Mosquitoes tho

Bees are already dying in great numbers.

Allowing the mass scale production and deployment of chemicals that kill bees really is the real life version of feeding crops with Brawno in Idiocracy.
Reminder that "bees" doesn't just mean honeybees.
Honeybees are actually bad for most place, they out compete native pollinators and fuck up ecosystems.
Is there another pollinator that could take their place?
- Not really. Flies do a surprising amount of pollination but they are not interested in the same things that bees are, or as diligent.
- Bats are responsible for a lot of pollination too.
  
  Most of our essential crops are wind pollinated (e.g. rice, wheat, corn) or asexual (potatoes). Colony collapse disorder is a real problem of course and colony management/replacement really eats into profitability, but domesticated bees won't become extinct because we so intensively manage them - however other species may be in real trouble. In addition, if we remove the main human causes of colony collapse (neonicotinoids?), they can potentially recover quite quickly, so it's not a problem that takes hundreds or thousands of years to fix, like some others - if we have the will to do it.

I'm gonna be different and say spiders.

Spiders are said to eat the mass of the entire human species every year. That's mostly insects. And that's an enormous amount to hand the insect population swell by every year. Of course, it would only increase that much in the first year. After that the growth would probably be exponential, at least until food sources were eliminated. Then the species that depend on those would go the same way, and so on up the food chain. It would probably actually be quite fascinating to see from an external perspective, or to study after the fact.

I'm not a biologist, but I feel like krill would be a strong contender, considering what percentage of ocean creatures rely on them for food.

This is an interesting take. I haven’t heard krill before.

Are we being pedantic about the definition of species? Mosquitos from the Anopheles genus (and only those species) spread malaria. They’re humanity’s #1 killer.

Driving them and the other mosquito species that spread human disease (Aedes spread dengue, yellow fever, Zika, and chikungunya) should be seriously considered.

Bees, plankton, and our keystone microbiota species.

Ants

The first thing I thought about seeing this post was the scene in Peacemaker where Vigilante talks about how Ants are more important than fish.
- idk what they do, but with so many its bound to be important.
- I’d honestly be worried about the sheer weight of all the ants disappearing at once affecting the earth.

The cave lion was our main predator, thanks to its extinction we went from prey to apex predator

It will be an invertebrate

Bees have to be the clear choice here. Butterflies going away would be terrible also.

@MadCybertist phytoplankton

Isn’t that a plant?

Along with the other good replies here I'll chuck in flies. They're the jack-of-all-trades-master-of-none in the insect world, sure, but they're the very bottom of sooooooo many food chains.

Plus we'd be buried in rotting material without them.

Sea urchins

Many people here seem to suggest the likes of plankton, worms, trees, flies, bees or butterflies. But these are not just one species, they're huge groups of organisms. It is interesting to think about extinction of whole groups, but it's necessary to understand it's on the same level as thinking about the extinction of all rodents or even mammals.