Wordorigins.org

The sense of deadline meaning the time when something is due is originally American newspaper jargon and first appears in 1920. From the Chicago Herald & Examiner of 2 January of that year:

Corinne Griffith...is working on "Deadline at Eleven", the newspaper play.

It’s so called because after that time the point of continuing to write the story is moot or “dead” as it won’t be published.

There is a somewhat different and slightly older printing sense that may have influenced the time-limit sense. A deadline is a line marked on the bed of a printing press that delimits the page and type that is placed outside the deadline will not appear on the page. From F.S. Henry’s Printing for School & Shop of 1917:

If the chase is one that just fits the bed of the press, make certain that the type does not come outside of the dead-line on the press.

And there is an even older sense of deadline that dates to the American Civil War, that of a line drawn around a military prison outside of which a prisoner may be shot. From the Congressional Record of 12 January 1864:

The “dead line,” beyond which the prisoners are not allowed to pass.

(Source: Oxford English Dictionary, 2nd Edition)