First consider a silly example. It was once proposed that dinosaurs became extinct because they got too big to have sex. This may seem reasonable at first; try to imagine a giant animal trying to mount another animal and ending up crushing it’s mate or simply being unable to get up that high ().
The reason that doesn’t happen is that natural selection would automatically eliminate any animals that were too big to copulate before they became common.
People have made similar arguments about peacock tails becoming so big that peacocks couldn’t avoid being eaten by tigers. Again, natural selection would prevent peacocks having ridiculously large tails from ever becoming common.
Now consider a more interesting example. This is an Osage-orange fruit, sometimes called the pawpaw ().
It’s been becoming less common since the ice ages because the animals that naturally ate it and spread it’s fruit, mammoths, ground sloths, and other megafauna died out when the ice ages ended. Without the help of those animals, the fruit often fell to the ground and germinated right under it’s parent tree instead of getting dispersed.
The Osage-orange tree evolved to become dependent on animals that eventually died out (). You could say that evolution nearly caused that tree to become extinct.
Mammoths provide another example of evolution leading to extinction. They persisted on Wrangel Island long after they had died out elsewhere; some were present there as late as 4,000 years ago.
The problem was that Wrangel Island isn’t big enough to maintain a very large population of huge animals; there were only about 300 individuals. Over time, mutations accumulated and the small population size made it difficult for natural selection to eliminate those mutations. The species gradually became less fit and eventually died out ().
So, mutations and the failure of natural selection to remove them, led to the extinction of mammoths on Wrangel Island.