Good question. Evolution will and does continue. Mass extinctions have happened in the past and in fact we are the product and beneficiaries of that process; when niches open up from extinctions, other species just move in and take over, eventually they will adapt to take the place of those who have left. Of course that's a useful perspective since it will happen. A large number of animals and plants are endangered and while the impact overall can't be estimated since we have never counted all the animals and plants, much less microscopic species, an example of our dire straits can be gleaned from the fact that perhaps half of amphibians are endangered,
There are various arguments that there is lost knowledge - when these animals and plants we have never even cataloged disappear we will lose hundreds of millions of years of biological 'experience', of lost knowledge as well as an aesthetic/ethical argument that this is clearly wrong somehow; like a mass murder of animals and plants.
I think that most people who have never seen these animals and plants, understood how wonderful and awe-inspiring they are, how many odd abilities that evolution has gifted them. It seems like a nearly universal experience that those who have seen this are pretty strongly in favor of efforts to save them.
But that's not the question at hand; really won't the ecosphere heal and eventually be okay? Now that we have discovered life so far under the ocean and in the depths of the earth that even nuclear holocaust would probably not touch them. The answer is yes. The ecosphere and life will go on. There are two other useful questions though:
1) will we go on with it? Mass extinctions are the result of changes in the environment - all living things in that environment will sink or swim as it were and humans should not exclude ourselves. Will the future human race deal with loss of oxygen content, the loss of fresh water and the misery of population explosion? If the environment just keeps getting worse, will we do well? We're all assuming that we will figure out how to feed ourselves, find drinking water when both food and water are not really growing to meet our growing population, not to mention the desertification of arable land and the rise in temperature that will impact human geography and economics over the next 100 years or so. If the environment were to stabilize at least we would not be standing on shifting ground. If you are familiar with Malthus' argument, when there is no food or land left to exploit nature will take its course with us - war, famine etc etc. That is also a natural state, which we've managed to slowly eliminate. We may be bringing it back in.
2) How long do we want to wait for the ecosystem to heal? After the mass extinctions of the past, hundreds of millions of years were needed for the species around to re-establish themselves as they are now. Is that really a good trade off for a few decades of industrial-economic expansion?
Until then, the survivors are not necessarily the ones we'd prefer.
I volunteered at a survey of shellfish in the San Franscisco Bay. At one time the shellfish here were so plentiful that the natives piled giant shellmounds the size of small hills from their regular diets. Now because of fertilizer runoff and pollution these shellfish are nearly gone - we didn't see one the entire day looking around the survey area. Instead they nesting shells were covered with a slimy bunch of sea squirts - plant-like animals which can grow quickly enough to absorb the nutrients and robust enough to ignore the pollution and chemicals. I'd probably rather have a bay scallops on my plate rather than a salad of seq squirts. The Bay is swimmable, but is plenty slimy.
Got to sign off now, but that's the outline of an answer...