One of the things people expect to happen as the global climate continues to warm is that species will die out in some areas (or entirely), but migrate into new areas, as warming makes it possible (or necessary) for them to start living in different locations. One of the likely drawbacks for people living in northern areas is that a lot of insects, including a number that spread human diseases, get killed off by cold winters, and if we end up with shorter cold seasons, we may have more problematic insect populations than we’ve been accustomed to. Global warming is also given as a reason for the increased spread of bark beetles in many forests in the US and Canada; the drought associated with that stresses the forests out, and makes them more susceptible to the beetles.
The infamous melting of the Arctic has been having a fairly well-publicized impact on the health of polar bears, who are running low on the sea ice they’ve previously depended on to get to the kinds of foods that sustain them best. It is too early to tell what the long-term impact is, or how often this will happen, but global warming has apparently brought a species of dolphin into polar bear territory, and the bears have been seen eating these dolphins for the first time ever. It could wind up that while warming climate screws up one food source for the bears, it ends up providing a new alternative; what the overall impact on the dolphin population will be, I don’t know.
Nature doesn’t like to leave niches unfilled, but it will take many decades (or centuries) to see what the “new normal” is going to be.