I am not an ornithologist, but I would hesitate to expect anything to be useful that hasn't been essential for millions of years. In general, though, it is fair to assume that (flying) birds are likely to do well during drastic global changes because they're already migratory and capable of following the climate.
Doing some very quick reading, the penguins are perhaps a useful case study. Estimates range on the clade being anywhere from 70 to 100 million years old, at a time and place which would have been quite a lot warmer than today, but they seem to have favoured the coldest reaches of the planet back then, as well. Although there are tropical penguins (as far north as Galapagos, above the equator!) they only arrived much more recently, perhaps as little as 4 million years ago.
So, with that one piece of evidence, while birds may be pretty good at getting away from the heat, they probably won't be exceptional at colonizing extreme environments. When biologists talk about organisms "remembering" conditions their ancestors experienced, they're usually referring to plant epigenetics, which, in computer nerd terms, is like saving a config file full of auto-calibrated settings for the next generation. Plants that have been exposed to certain soil or weather conditions will tend to produce seeds that are more resilient to those conditions, but their genes haven't changed, just a few pieces of metadata that tip the scales toward better adaptation. It's likely that these can encode short-term weather cycles (e.g. El Nino years) but long, irregular trends (much less short, sudden, anthropogenic ones) could not possibly constitute a selective pressure to drive the evolution of a specific memory-like mechanism; at best the plants of each new generation will have to spend some effort adapting directly. (Fortunately, a single field of corn produces an absurd number of offspring. Plants are very good at exploring the problem space of evolution!)
If you see any other mention of "genetic memory," that's a misunderstood reference to Frank Herbert's Dune, and it is not real science. There are cases where the body alters its own genome for practical purposes—the immune system creates new antibodies by randomly deleting blocks of code from a master template protein sequence, and there's some evidence neurons use their own DNA as a limited form of data storage—but nothing substantial is heritable. Heritable epigenetic phenomena do exist in animals as well, but they're probably not very interesting and seem almost vestigial or dysfunctional.