There are at least four books on my office shelf that deal with the impacts of climate change on biodiversity. It is a huge area.
In short, many species are undergoing latitudinal range shifts (eg. they now occur where previously they did not) which is a response to seeking more appropriate climate regimes. We have evidence that some species are undergoing altitudinal range shifts as well, thus influences the ecosystems they enter and raising questions as to where species can move to if appropriate resources are not available where the future suitable climate is going to occur.
Some of the first signs we had were changes in phenology, eg. the timing of events (some frogs are breeding 1 month earlier than they were early last century), but it includes earlier flowering, emergence of insects, first bird calls etc…
The problem with this is that some species are changing their timing of events due to climate change, but others have reproductive or emergence triggers associated with photoperiod (day length) which doesn’t change. So some birds are arriving at breeding grounds after or before the emergence of prey items, thereby altering the availability of food resources.
We know that many corals are under stress and growing slower, possibly due to thermal stress from rising water temperature, but more of a worry is increasing oceanic acidity due to a chemical reaction with increasing atmospheric CO2. This may affect many ocean species which use calcium carbonate as a building material. This is a very serious issue that few in the public are aware of and is based on a simple chemical reaction that has lowered the pH of the ocean by 0.1 over the last decades.
The only species I am aware of whose extinction is strongly linked to climate change is the Golden Toad. Dozens of harlequin frogs have gone extinct, also possibly linked to a combination of climate change driven increased disease exposure in the form of a chytrid fungus skin infection.
Large drops in some penguin colonies due to declines in sea ice associated krill colonies. Several polar bear populations are declining, also linked to climate change (which lead to their recent change in conservaiton status).
Anyways, you get the picture. Multiple impacts, some are good for some species, many threatened taxa are expected to do worse under climate change. Depends on the species considered to be honest, whether or not climate change will favour them, disadvantage them or in some cases lead to extinction.
If you’re really interested I’ve attached a link to one of the best scientific reviews available, which is a few years old, but gives a very thorough picture of the processes that are going on.
http://cns.utexas.edu/communications/File/AnnRev_CCimpacts2006.pdf