I'd say they became obsolete in the late 1970's/ early 1980's for 2 main reasons:
- Radios on trains became more reliable and almost universal. And, the elimination of cabooses and the down-sizing of crews to just 2 or 3 men put them all in the cab of the loco, so there was only 1 point of contact needed for the flimsies any more. Plus, the conductor was also there - and the best person - to copy down the track warrant. In effect, the conductor became a mobile, on-the-spot, real-time, 24 x 7 operator.
- The widespread closing and elimination of train order stations and their operators. Also - and as part of that, the railroad's 'command, control, communications, and signals' systems and culture gradually became more comfortable with radio-issued orders that were written and checked by and between just the dispatcher and the train crew, and did not need the involvement and records kept of that transaction by yet another person, the lineside operator.
I believe ConRail was one of the first to use radio-issued track warrants on a wide-spread basis, but maybe that's just because I was most familiar with it at the time that change-over was occurring.
- Paul North.
P.S. - EDIT: This "ABC's of Railroading" article on "Train Orders" says 1986:
http://trn.trains.com/en/Railroad%20Reference/ABCs%20of% 20Railroading/2006/05/Train%20orders.aspx
See also this one on "Railroad's Traffic Control Systems" at:
http://trn.trains.com/Railroad%20Reference/ABCs%20of%20Railroading/2006/05/Railroads%20traffic%20control%20systems.aspx
"This Fascinating Railroad Business"
(title of 1943 book by Robert Selph Henry of the AAR)