History is fact littered with civilisations engaged in unsustainable practices. Some of the worst offenders have long since collapsed, but the ecological damage they caused or contributed to often still have reverberations today.
Prior to the industrial revolution, agriculture dominated human life. Accordingly many examples of unsustainable ancient cultures are related to agriculture. Perhaps the most famous case is that of Ancient Sumer, in what is now Iraq. For millennia the Sumerians were able to thrive by harnessing the Euphrates and the Tigris for irrigation. But the very technology behind Sumer's rise also eventually brought it down: the rapid evaporation in this arid region and the lack of drainage led to a gradual accumulation of salt on Sumerian farms. The rising soil salinity eventually destroyed Mesopotamia's agricultural productivity, and with it the Sumerian civilisation.
After 1,000 to 5,000 years of successful irrigated agriculture, the Sumerian civilization declined. Numerous references to canal building in Sumeria from the third millennium B.C. are available, but no record of drainage canals being built to sustain agriculture exists.
Tanji, K. K. "Agricultural Salinity Assessment and Nanagement." New York: ASE. 1990.
Another example from classical antiquity are the Garamantes, who once dwelt in what is now southern Libya. They created a flourishing society in the rather inhospitable region, by mining fossil water for irrigation. Using a system of underground tunnels known as foggara, constructed with immense slave labour, they were able to tap into a vast underground lake formed in the Sahara's wet, distant past. This allowed them to engage in the large scale cultivation of Mediterranean crops even as desertification swept past their settlements.
For over a millennium the Garamantes thrived and dominated perhaps up to the whole of Fezzan, outlasting even the Western Roman Empire. However, fossil water is, as you might guess from the name, hardly a renewable resource. The Garamantes were probably forced into a vicious cycle of acquiring ever more slaves to dig ever deeper tunnels to extract some more of the dwindling supply of water. Once the reservoirs were depleted, it was - and is - impossible to sustain large scale farming in their arid climate, and the Garamantian kingdom collapsed and passed out of history.
By the fourth century C.E., when slaves formed perhaps as much as 10 percent of the population, the groundwater table that sustained the Garamantes began an irreversible decline. Excavations to reach the aquifer grew more difficult and labor demanding. The fossil reserves proved finite, and around 500 C.E., the kingdom collapsed.
Carney, J. and Richard Nicholas Rosomoff. In the Shadow of Slavery: Africa’s Botanical Legacy in the Atlantic World. University of California Press, 2011.
The early Maori tribes of New Zealand offer a more recent example. Upon their arrival, the Maori found a virgin land plentiful with walking feasts. The large animals then extant in New Zealand, most famously the giant flightless moa birds (though seals were probably a greater part of their diet), had evolved in isolation and were easily hunted. They were annihilated within a couple of centuries by a combination of overhunting and habitat destruction. Though not as catastrophic as systematic agricultural failure, the loss of their main sources of meat protein rocked the nascent Maori world. Easy access to protein had fueled a population boom, which was now far more challenging to feed.
However, the Maori adapted. The mechanism of rāhui was developed alongside the general Polynesian concept of tapu, whereby certain resources or areas were declared taboo to allow nature time to regenerate. For example, the surviving seal population in Chatham became protected, their hunting limited to sustainable numbers. In this way Maori society were able to transition to sustainable development, and thereafter maintained a relatively healthy and slowly growing population until the time of European contact despite the restricted supply of proteins.
Several prehistorians allude to signs of increasing tapu from mid-prehistory . . . The Great mangaunu shark-fishing enterprises were annually restricted to two days [and] other resources also had restricted-use seasons, enforced by tapu and rahui, and converging precisely with prime productivity . . . This ideological shift [to a restricted and therefore sustainable exploitation of the environment] underwrote the change from an extractive to a sustainable economy.
Belich, James. Making Peoples. Penguin UK, 2007.