Dimmitt hasn't since 1976 (unless they snuck one in there recently and I missed it)
Northbrook (Spring Branch ISD) hasn't EVER made it since the school opened in 1973.
Diamond Hill-Jarvis (Fort Worth ISD) hasn't either, since 1954, from what I've read.
And, of course, the grand champion: North Dallas (Dallas ISD), which hasn't made the playoffs since 1952.
Unless you count six-man's Prairie Lea, which hasn't had a playoff appearance since 1944 - but they stopped playing football between 1950 and 2001. It's easy to have a 50+ year streak when you don't even play the sport!