In the particular photo you offer as an example, the person is doing two things: bugging his eyes and doing some version of duck lips or a duck face. Here is the entry for bug-eyed in The Random House Historical Dictionary of American Slang (1994):
bug-eyed adj. having staring eyes, as from fear or astonishment
[First citation:] 1922 in OED2.
Duck face is a much more recent slang term, dating to 2006, according to the extensive history of the term at Know Your Meme. That site defines duck face as
a pejorative term for a facial expression made by pressing one's lips together into the shape of a duck's bill.
There are, of course, more and less exaggerated forms of this expression, and some variations have their own slang names. For example, Know Your Meme also has an entry for sparrow face (evidently a translation of the original Japanese チュン顔 or ちゅん顔, which the site says would more literally be rendered as "chirp face"), which it dates to February 2013 and defines as
a facial expression made by slightly parting one’s lips with eyes wide open.
Technically, I would say that the guy in the OP's photo is doing a manic sparrow face, rather than a duck face. The effect of the combined positioning of eyes and lips is to produce a look that is only slightly more exaggerated than the one that male model Derek Zoolander calls "Blue Steel" (or "Ferrari" or "The Tigre") in the movie Zoolander.