A difficult question is whether an ecosystem is alive in the sense of a single living organism. Any form of life has a form of homeostasis which keeps it at maximum entropy, that is, shedding heat. Meanwhile, its internal life processes are negentropic, increasing the degree of order of matter in the body. This, along with the ability to reproduce, are usually cited as the two most important attributes of any life form.
An ecosystem typically does not reproduce, and it typically has a homeorhetic equilibrium and not a full homeostasis. However, a virus or prion or molecular assembler or perhaps a clanking replicator also lacks their own reproduction, homeostasis, and thus full status as a life form, but it would not be difficult to identify when they had been "killed", i.e. lost their integrity or "died", unable to perform their normal functions any more.
From a strictly human point of view, the most reasonable way to assess the life or lack of life of any ecosystem is whether it is capable of an ecological yield of fresh air, clean water, etc. - providing nature's services, on which our own lives depend. This is the reason why we might care about ecocide, of course, which aside from being an aesthetic or ethical issue, is of course a matter of sheer survival for those who depend on the endangered ecosystem in any way.
From this point of view, there are many known cases of utter disabling of nature's services due to pollution, erosion, loss of topsoil due to strip mining or overintensive agriculture - perhaps leaving a minimal ecosystem such as a desert not capable of performing the same services or life forms as prior to the ecocide, flooding, salt intrusion - such as due to shrimp farming, and (most commonly) deforestation.
Some well known examples are the drying up of the Aral Sea, the advancing of the Sahara Desert into what were the grain-producing regions of the Roman Empire, the erasure and poisoning of farm lands along the Western Front during World War I (see technological advancement during World War I for details re: mud) and the total loss of all trees from Easter Island, which today is simply grassland, incapable of supporting a large population. This last is a signal case often cited by ecologists, who refer to Easter Island Syndrome as evidence that human beings may be naturally inclined to ecocide, and do indeed shit where they live, as the popular expression (and the song Humans Are Stupid, by Mendelsohn Joe) suggests. See also misanthropology.
The related concept of dieoff in population biology refers to the precipitous drop in any population once it has overgrazed its environment to the point where it is simply no longer viable as a population. This usually occurs far before a full-spectrum ecocide, however, as few organisms consume a wide range of foods, nor directly transform (or "terraform") ecoregions simply to serve their own purposes. Thus total extinction of all life seems unlikely.
Except, of course, where humans are involved. In that case, anything would seem possible.
See also: list of ecology topics