Joe Gibbs, the former Washington Redskins coach (1981–1992, 2004-2007), is widely credited with making the phrase popular in a Washington Post article in 1982. Supposedly, he tried to use it to motivate his 1981 team that was off to a terrible 0-4 start and had the worst offensive efficiency in the league when inside their opponents' 20 yard line. The concept behind the phrase was created by a statistician (Dave Plati) working for the Denver Broncos in 1980, in which he was given free reign to create new statistics, so he began tracking offensive efficiency inside the opponent's 20 yard line.
Joe Theissman, the famed Washington Redskins quarterback from the early 80's, remarks in his book The Complete Idiot's Guide to Football:
"The red zone was so named because it represents the area where a red flashing light should go off. Don Shula, long-time NFL coach, called it the green zone because that was where the money was made.
However, it was originally a military phrase used for an area that was particularly dangerous, i.e., an area within firing range of the enemy that had no cover. The first documented use I can find of the phrase was during WW1, when areas of France were so devastated that they were deemed uninhabitable by humans and labeled "zone rouge" (which means "red zone" in English). They often contained unexploded ordinance, toxic chemicals, dangerous gasses, and many human remains, and the French government passed laws to temporarily (or permanently in some cases) keep humans from inhabiting or building on the areas.
Sources: Football use - 1,2,3 - Military use - 1