There would be no more aphids, mustang horses, fire ants, lady bugs, Japanese beetles, and honeybees in North America!   It's hard to imagine how much stuff would grow here without those dang aphids.  They suck the life out of young growing plants.  And those Japanese Beetles do so much damage to the east coast of NA.

The worst example I can think of is one of the now uninhabited Hawaiian Islands that let rats loose and they ate so much of the trees (Sandalwood) that storms were able to wipe out enough of the island that people can't live there anymore. 

I love bunny rabbits, but I get that Australia doesn’t benefit from all the wild ones.

 To answer the orginal question - sometimes, but you have to be very selective.  The results can be devistating.  Like how they killed all the wolves in Yellowstone, and if they had not put them back all the forest would have gone.  All the beavers would be gone.   Which would cause massive floodings downstream - which is much of the USA.

I'm totally ok with it.

Yea some of those species are harsh; especially the Japanese beetles. But we can keep the honeybees; I like them! I meant invasive as in having a negative impact on an ecosystem, and not necessarily just introduced

Yea, I think we would proably keep honeybees in North and South America.  There are some natives ones that make some honey, but Honey Bees do it best!

Although, I wonder if their is some ant nector or something we haven't tried yet that is delicous.

Have you tried local honey from a farmer's market?

Also, I have a hive of bees try to move into my apartment.  I was like what's that noise?  Then saw the swarm.  That was an excieting day.

No, we definitely should not drive other species to extinction. That is unethical.

However, rendering ourselves unavailable as prey or inhospitable as hosts/reservoirs is ethical. If a species goes extinct because it can no longer infect or survive in our bodies, that is the natural consequence of evolution. This scenario is very different from genetically altering inconvenient species or hunting them to total eradication.

If the ends are to be met and are the same, it is better to reach them through ethical courses of action.

Yep I have! It is very delicious. Never had a bee problem but wasps have always tried to move into my buildings haha

Kill them all. Bees are fine tho.

The Fourth Doctor once said that if he killed off all of the Daleks that he would be just like them,

Now there's a war of attrition.

It's suspected that the Rabbit is the single greatest factor in the extinction of native species since European colonization in Australia; bigger then humans, or even feral cats/dogs/foxes. The land damage they've caused through erosion is projected to take centuries to repair.

Here down under we've delibrately introduced not one but two foreign viruses to try to wipe them out; first Myxoma in 1950, then Calici in the 90s. Both brought numbers plummetting down for a while, until they acquired a resistance and rebounded.

Hell, we even built the world's longest fence to try to contain them; 3256km.

Their extermination at least from the continent would be hugely beneficial to most other living things there, including people.

That's one of the thigns we learned from HIV/AIDS treatments.  You always need 3.  Even if the 3rd one is only like 60% effective. 

Plus now half of the bunnies are immute to Myxoma.  So it's not so effective anymore.

I do know they were releasing something very recently in Sidney - from a bunny list.  So maybe there is something new going out.  Rabbit should only be indoor pets in Australia - not pest.

Or they could make tiny bunny killing robots.  They also need to do that with the monguses in Hawaii.  Similar devistation.