Well, probably, some will, given enough time. But that might be a problem.
The world is changing fast. So the question should really be 'Will the earth's organisms be able to adapt to the changing circumstances fast enough?'
To test this, researchers at McGill University decided to track the fate of more than 2000 populations of yeast (more specifically baker's yeast, or Saccharomyces cerevisiae). Why yeast? Two reasons: 1) it's a well-established model organism in biological research, so a lot is known about its genetic makeup, and 2) it reproduces in a matter of hours.
Figure 1: Saccharomyces cerevisiae, or baker's yeast (Source: Wikimedia Commons, Masur)
By using a robot, they were able to monitor and manipulate the environmental stress (in the form of salt) of the populations to varying degrees. This allowed them to study evolutionary rescue, which is the ability of a population to adapt fast enough to changing circumstances in order to survive.
They found that there were three important factors in their experiment, affecting the likelihood of evolutionary rescue.
1) The severity and rate of change in the environment. The more gradual the change, the better.
2) Prior exposure to the stressor (in this case, salt).
3) The degree of isolation from other populations. Contact with populations that had already experienced environmental change increased the survival of another population during a rapid and severe perturbation.
After 50 - 100 generations, adaptations to the new circumstances could be noticed, which is surprisingly fast. Doing a similar test on mammals would be interesting, but would require a lot more time. Probably too much time given the current rate of environmental change.
The next best thing is to develop a model and extrapolate from it.
Bell, G. and Gonzalez, A. (2011). Adaptation and Evolutionary Rescue in Metapopulations Experiencing Environmental Deterioration. Science. 332(6035), pp. 1327 - 1330. doi:10.1126/science.1203105.