In Alaska, one of the famous examples is the Steller Sea Cow, named after Georg Steller, the naturalist who sailed aboard Vitus Bering's expedition to North America in the 1740s. Steller documented many animals and plants for the first time in the European record, and as such, his name shows up quite a bit if you're studying the natural life of the North Pacific.
The sea cow went extinct not long after it was discovered by Bering's sailors. A manatee-like creature, it had ranged across the North Pacific but by 1741 when Steller observed it, it was found only in a single isolated population in the Commander Islands. It is thought that Native hunting exterminated it elsewhere, and Russian hunting was the final blow.
The Russians and other Europeans who first sailed to Alaska frequently found themselves short of food. Scurvy was common. In their logs and journals, the lack of beef, bread and fresh vegetables are frequently mentioned. The Steller, slow and easily hunted, was the closest thing to beef for the hungry Russians sailing between Okhotsk and Alaska. Within 30 years of its first recording by Steller, it was extinct.
Dodos were made extinct partly due to over-hunting by sailors, partly because they were ground-nesters do their eggs were easily eaten by ships' dogs and partly because the stopping ships introduced rats to their islands.
It is possible that silphium, prized in the ancient world as a medicinal herb and possible birth control/abortifacient, went extinct due to human over-consumption. (But is is also possible that it was related to the major ecological shift in northern Africa that changed a fertile farmland into part of the Sahara desert.)
The horse, despite evolving in the Americas, went extinct on those continents not long after humans arrived there. It survived in Asian populations, but some speculate that the horse, along with other mega fauna of North and South America, was hunted to extinction.
It seems that climate change could also be a factor in these extinction events, impossible to know "for sure" if they might have survived one shock (heavy human consumption) if not for another (rapid change in environment.)
One rather big candidate is the Moa which was found only in New Zealand and hunted into extinction in a rather short time span after the arrival of Maoris.
For me the most astonishing case is that of the passenger pigeon of North America. Before the arrival of Europeans it is estimated that there were 3-5 billion. Early colonialists describe swarms that could truly darken the sky and even later (maybe with a bit of exaggeration)
One flock in 1866 in southern Ontario was described as being 1 mi (1.5 km) wide and 300 mi (500 km) long, took 14 hours to pass, and held in excess of 3.5 billion birds. That number, if accurate, would likely represent a large fraction of the entire population at the time
50 years later there were none left.
It should be mentioned that both animals mentioned here also suffered extreme loss of habitat in a very short time frame and in the case of the passenger pigeon they were also hunted for other reasons than food.
Many animals were hunted to extinction for food. A major example would be the various species of moa, which were large, flightless birds found on the New Zealand islands. I believe fossil analysis can date the birds' extinction to around the 17th century. Another case would be the great auk, which suffered from egg-harvesting as well as being killed for its meat and feathers. One of the last living specimens was strangled on St. Kilda early in the 19th century because it was believed to be a witch.
However, there are usually a combination of factors other than only hunting for food. In the case of Steller's Sea Cow, the population was small when it was discovered. Often habitat loss is to blame as well.
Sources: My own memory, partially (I have a degree in Bio). petermaas.nl seems to be a very trustworthy and well-researched website on extinctions. Unfortunately ARKive's section on extinct animals is not yet complete.
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