Nothing happens. After all, many, many scientific papers are wrong, and many of them aren't recognized as such for a long time. It's often not fair to criticize the prizewinners this way either, since they can be very careful and competent scientists, they just weren't "lucky" enough for their life's work to be correct.
Perhaps the most clear-cut example hearkens all the way back to 1926, when Johannes Fibiger won the Nobel Prize in Medicine for "for his discovery of the Spiroptera carcinoma." In layman's terms, he found a tiny parasitic worm that causes cancer. Subsequent research conducted in the decades following his receipt of the award would show that though the worm definitely existed, its cancer-causing abilities were entirely nonexistent. So where did Fibiger go wrong?
Though widely respected and considered to be a careful and cautious researcher, Fibiger fell victim to improper controls and inadequate technology. To elucidate his hypothesized connection between parasites and gastric cancer in rodents, he fed mice and rats cockroaches infested with parasitic worms and observed what he thought were tumors grow inside the rodents' stomachs. Later studies would show that they were not tumors but lesions likely caused by vitamin A deficiency, which resulted from a poor diet.
It's hard to fault Fibiger or the Nobel Committee too much for this blunder. At the time, cancer was much, much more of a mystery than it is today, and Fibiger worked tirelessly to solve it, exploring all sorts of hypotheses, not just those involving parasites.
That said, I wouldn't put much weight into the argument that dark matter/dark energy don't exist. There've been many attempts to do away with one or either ever since the 1980s. Every now and then they make the media too. But standard dark matter theory has scored many significant successes to the point that Sean Carroll said in 2012 that "by now we’ve accumulated enough data to conclude that the universe cannot be explained solely by modifying gravity". The evidence for dark energy is less compelling, but still getting more compelling constantly. Until we know what they are people absolutely should work on alternatives to both of them! However, these papers come with big caveats about how likely they are to be wrong, which the media seldom mentions when they write about them. For example, the paper cited by the article you link dates to 2017, which is ample time for the community to process it, but Lambda-CDM remains the gold standard (i.e. the paper hasn't convinced the community and is most likely wrong).