The first appearance of all-white eyes in general is probably Little Orphan Annie, which has had characters drawn with blank white eyes at least since 1934:
This seems to have been a conscious choice on the part of creator Harold Gray; the earliest (1924) strips don't have this feature:
Unfortunately (or fortunately), I can't find any information on why Gray changed to this more-than-slightly unsettling style.
The earliest version of the white-eyed mask is The Phantom; he's been rocking it at least since November 1936:
This is actually The Phantom's second newspaper story, "The Sky Band"; the first began in February 1936 with "The Singh Brotherhood", but I've been unable to find scans that clearly show his eyes. The nearest I can find is the following panel, from "The Singh Brotherhood" but of indeterminate date, which faintly shows his irisless eyes:
According to Wikipedia, Phantom creator Lee Falk revealed in an interview that the white eyes were inspired by Greek busts:
In the A&E American cable TV documentary The Phantom: Comic Strip Crusader, Falk explained Greek busts inspired the idea of the not showing the Phantom's pupils when he was wearing his mask. He (incorrectly) believed that Ancient Greek busts displayed no pupils (they would have been painted on originally, which over time faded) which he felt gave them an inhuman, awe-inspiring appearance.