Last team to win after being down 3-1 was the Suns over the Lakers in 2006. Suns had HCA and game 7 was the infamous one where Kobe refused to shoot the ball in the second half to prove the Lakers couldn't win without him.
The last time before that was in 2003 when 8th Seed Orlando led by TMac was up 3-1 against 1-seed Detroit. After game 4 and a 3-1 lead, TMac infamously said "It feels so great to finally get out of the first round."
Time before that was the 97 Eastern Semis between the Knicks and Heat. Worth Watching. Sage Rosenfels isn't the only QB to do a Helicopter. Charlie Ward did when PJ Brown threw him over a cameraman.
https://www.youtube.com/watch?v=5C60dX7m89Y&t=3m09s Knicks had suspensions to several key players including Patrick Ewing and John Starks.
Before that was the Rockets-Suns in 95.
Then you have to jump back to 81 when the Celtics beat Dr. J in 7 after being down 3-1 enroute to Bird's first title.
That is every single 3-1 comeback to win in 7 since the beginning of the 3-point line (in my mind, the dividing line between ancient NBA and Modern NBA.)