This question is really hard to answer because it’s so general. There were so many different groups in North and South America over such a long span of time that, obviously, there were major differences in how groups and individuals conserved or did not conserve the environment. But by and large, prior to there being real pressure on resources (i.e., an influx of European settlers competing for land, integration into Western commodity markets), North American natives were not particularly conservationist. In fact, some were quite wasteful. One example is the buffalo drives, where one method of hunting buffalo was to cause the herd to stampede over a cliff. You can’t stop a stampede once it’s started, and this lead to a lot of wasted meat as the hunters only took what they wanted.
One of the most commonly cited examples of ecological damage caused by northeastern Native Americans is the beaver. This animal was a crucial part of the northeastern ecosystem, keeping rivers and forests healthy, but they were completely and quite eagerly hunted out. Some groups and individuals belatedly saw over-hunting as harmful, but there was nothing intrinsic in northeastern cultures that prevented it.
The biggest environmental impact that’s often quite overlooked is due to fire. This can be taken either way as good or bad. Europeans first arriving in North America remarked on how park-like the forests were; park-like meaning they had a clear understory. This was due to frequent, deliberate burning for agriculture and hunting, with the result of actually helping to prevent larger forest fires by keeping fuel out of the understory. They also gathered trees for fire wood, and like any other larger settlements cut so much close to settlements that wood became scarce in some areas.
Part of the widespread association with “Indians” and “environmentalism” has to do with the environmental movement in the 1960s and 70s. See for example the Crying Indian commercials. The origins of the idea are much older than that though, and stem from the romanticization of Native Americans as “Noble Savages” who are more a part of nature than Europeans. A book that talks about this idea is Ter Ellingston’s The Myth of the Noble Savage, and he dates this connection to the mid-nineteenth century.
There are two really good books on this subject that I recommend. The first is William Cronon’s Changes in the Land: Indians, Colonists, and the Ecology of New England. His focus is mostly on English colonists, but he also looks at commodified game animals (i.e., beaver) and the role of wampum in the early colonial economy. He also writes about the fur trade in connection with the spread of epidemic diseases and looks at the effects of Native settlement collapse on forest landscapes (they grew more “foresty” and less “park-like”).
The other book directly speaks to your question. Shepherd Krech’s The Ecological Indian: Myth and History asks whether or not North American Natives were historically conservationist or “ecological.” He looks at some of the bigger Native American groups, like the Hohokam who had pretty massive irrigation networks until something happened and the communities collapsed. He also gets more into the environmental history “theory” as it were, which challenges the idea that human societies are somehow not a part of what we call “nature” so that they can actually have some kind of impact on it. He also looks at things like the buffalo hunts I mentioned, the Pleistocene extinctions of large mammals, and the deer and beaver hunting in the eastern part of the continent.