In pushing other species to extinction, humanity is busy sawing off the limb on which it perches.

Paul Ehrlich (b. 1932) American conservation biologist and ecologist

(Attributed)

All citations for this I found are from a reference in Elizabeth Kolbert, The Sixth Extinction: An Unnatural History, ch. 13 (2014), to a sign in the American Museum of Natural History's Hall of Biodiversity which "offers a quote from the Stanford ecologist Paul Ehrlich," giving the above text.

I was unable to find the phrase in Ehrlich's written work, though it could be from a speech, media comment, etc.

In Ehrlich's One with Ninevah: Politics, Consumption, and the Human Future (2005), the epigraph for chapter 2 is a quotation from William R. Catton, Jr., Overshoot: The Ecological Basis of Revolutionary Change, ch. 2 (1980), regarding Earth's finite non-renewable resources:

This fact puts mankind out on a limb which the activities of modern life are busily sawing off.

This might be the source of a misattribution to Ehrlich, though the context is not quite the same, and the metaphor of sawing off the branch one is sitting on is not unique to Ehrlich or Cotton. More research is needed.