Data obtained from the Stack Exchange API is cc-wiki licensed.
This license explicitly encourages the user of the API to copy, redistribute, remix and transform the data obtained from the API for any purpose, provided that appropriate credit is given and that the cc-wiki license remains.
If data obtained via the API was used to build a new website that displays tens of thousands of unaltered questions and answers reformatted for easier browsing, and if one of those questions contains a link to a copyrighted image, then would the user of the API be liable for displaying that image link on the new website?
To be clear, the data obtained from the API contains just a link to an image hosted on stack.imgur.com. The image was uploaded there by a Stack Overflow user who did not have a license to do so. That user is clearly liable; I'm asking whether the user of the API is liable. The data that they obtained via the API may contain hundreds of thousands of image links, any of which may violate copyright, and this places a burden on them to verify all image links for potential copyright violations before sharing and remixing the data.
In light of the top answer below, I've discovered a way to make easy money. Behold!
Copyright Infringing Poem
If the following poem is republished anywhere other than on law.stackexchange.com then I will demand $1000 in damages.
by Jason Hutchens
Oh mighty sausage, how juicy and sweet
The delicious smell of your sizzling meat