It's considered a slang so it isn't in the OED. But luckily Random House comes to our rescue.
It has the ordinary verb 'to file' being used to mean 'to throw away' since 1934.
1934 Weseen Dictionary of Slang 334.
File 13 doesn't appear til about 1941, with Random House giving it a Millitary origin, meaning 'a wastebasket'.
1941 AS (Oct.) 165: File. a waste basket. (It appears that they somehow left out the 13 bit).
Where the number 13 come from is anyone's guess. Perhaps it's nothing more than the next installment in the continuing saga of that particularly unlucky number.
Reply from Doug Gilbert (Chai Yi - Taiwan)