How could a worldwide increase in temperature affect organisms?
There really aren't enough specifics provided in this question for it to be answerable.
There really aren't enough specifics provided in this question for it to be answerable. Organisms include humans, trees, orchids, spiders, fleas, domestic dogs, killer whales, parrots, tuna, salamanders, bacteria and so forth.
To answer this question, we need to know how much of a temperature increase, over what period of time does that increase happen, and which organisms are of interest?
Some organisms may benefit from a rapid increase, some may go extinct, some may not be affected at all by a temperature increase of 1 degree celsius, and so forth.
Some generalizations we might be able to make (keeping in mind that these are generalizations) are that 1) species unable to adapt quickly to rapid changes are more likely to go extinct than those able to adapt, 2) as temperatures warm, body size decreases for endothermic (cold-blooded) animals and increases for ectothermic (warm-blooded) animals, 3) warming temperatures will transform the polar regions and lead to a loss of Arctic species.
Here's a table modified from the fifth IPCC report describing potential risks and adaptation issues due to warming temperatures:
Here are some studies on increases in temperature and the effect on an organism:
Hotter, drier weather linked to shrinking salamander size