If the alarmists are correct and peak fossil fuels is just around the corner, then after the peak the cost of fossil fuels will gradually go up until alternatives -- alternatives that we know about and can make use of relatively quickly -- make more economic sense. At some point, certain sources of oil will become economically viable which aren't now. At a higher point, it will make more sense to create synfuels, or switch to natural gas, or to build more solar or wind plants and all-electric cars. Will this be economically worse than just using cheap fossil fuels? Yes (assuming that the pricey energy doesn't motivate breakthrough technology such as fusion power or really good solar.) Will the world end or even change dramatically? No.
Heck, it's an insane, absurd speculation, but maybe at some point energy will become so expensive we'll even come to our senses, stop listening to Greenpeace, and build nuclear plants to use that 10,000 years of uranium. Weirder things have happened.