It starts with clear progress on the field. That has to be evident, or doubt begins to creep in. Dooley never had that, so after 3 he was gone. Butch had it, so he got a couple more years; but couldn't hold it, the progress became regress, and he was out.
But there are things we fans don't always see, sometimes only hear vague rumors of, that contribute as well. It's the intangibles that help the AD determine whether the program (not just the team, the entire program) is on track. Are the lads going to class, making good grades, staying out of trouble? Is the mood in the locker room positive, supportive, team-oriented? Are the coaches leading by example in their personal lives? These things also can give a coach another year, or take one away from him.
Dooley's habit of embarrassing his players publicly for a laugh (showers, Rommel, etc.), as well as the extreme extent to which he alienated both former players and high school coaches, probably cut off any good will that might have gotten him a fourth year. Butch, on the other hand, had the intangibles working in his favor in the early years--embracing Tennessee tradition, talking a good talk (though it would turn out he wasn't walking the walk). That turned around and hurt him after the questions began to pile up (did he hit one of his players? did he have to embarrass the program with that inane comment? was he making promises to recruits that he couldn't keep?).
At the end of the day, though, the team has to win. At a place with championship trophies in the cabinets and a whole lot of pride throughout the fan base, that means winning at the highest levels. Without that, time will eventually run out, no matter how well everything else is going.
p.s. One after-thought: almost goes without saying, but it's also a different world in the 21st Century. Johnny Majors would never have gotten 16 years with his roller-coaster pattern of progress. If he were coaching in this decade, he'd probably have been fired at the end of his fourth year, and certainly by the end of his sixth.