Reverse narratives can be fascinating when done right. Here are some music videos - in a loose order of preference - that I can remember which use this technique with mixed results. I've expanded this list to included all competently made "backwards" videos that I can remember / find as there aren't all that many out there.
Enigma - Return to Innocence (1994)
This was one of if not the first video I saw which used a reverse chronology. You can guess the theme of the video from the title of the song :)
The Pharcyde - Drop (1995)
Directed by Spike Jonze, this is supposedly the first "reverse music video" (even though the release dates don't compare favorably with the Enigma release). Rather than the song and the video, it's the making-of YouTube link (YT 2) which is the more interesting.
Cibo Matto - Sugar Water (1996)
Weird band, weird video, but intriguing nevertheless. It's not surprising considering that Sean Ono Lennon is involved in this band :D This MV is low budget but well made and uses a split-screen technique that .. meets in the middle. Nifty! (Anybody want to attempt explaining the storyline?)
God Lives Underwater - From your Mouth (1998)
No actual story and while the lyrics for the song allude to something else altogether, we get to see what eating looks like in reverse :) It's a single take featuring champion hot-dog eater Hirofumi Nakajima removing vast quantities of food from his mouth. It is directed by Francis Ford Coppola's son, Roman.
Coldplay - The Scientist (2002)
It's interesting to listen to the song first and dwell upon the lyrics prior to watching the video. There's a nice article elaborating on the storyline as well as the line of thought behind the project.
Jack Johnson — Sitting, Waiting, Wishing (2005)
Pretty run-of-the-mill. The linked video is a "restored" version.
Linkin Park - Bleed It Out (2007)
A pretty catchy song; the band plays normally while the audience is shown in reverse suggesting that the two entities were filmed separately. The audience bit also feels like a continuous shot and appears to be a meaning brawl until you get to the beginning/end. +1
Mutemath - Typical (2007)
We basically see the band horsing around on stage. But the effects, and the editing in particular, are nice.
Depeche Mode — Wrong (2009)
Rather surreal video from Depeche Mode which is open to multiple interpretations. Pay attention to the lyrics too. (Yes, it should possibly not be on this list. But that's kind of the point of the video, isn't it?)
Ricky Martin - Private Emotion (2009)
A stale song with a straightforward reverse narrative which is probably more about symbolism than anything else. (Also, this must have come out before Ricky did.)
David Cook - Come Back to Me (2010)
A reverse narrative for 95% of the video after which the clock literally starts moving forward again. There's nothing terribly innovative going on though.
Grouplove - Tongue Tied (2011)
The backwards video kicks in after about 20 seconds. It's rather dull, I'm afraid. The music is decent and reminds me of the band, Suede.
Peter Bjorn And John - Second Chance (2011)
A rocking song with a backwards video of the band, again, horsing around.
Yeah Yeah Yeahs - Sacrilege (2013)
This is a reverse narrative while not necessarily with the video being played backwards. It follows the story of a woman being burned alive and a man being shot. The narrative explains how this came about. It's rather well done with a lot of symbolism.
Leningrad - Tattoo Artist (2017)
This is a fun NSFW video to accompany a fun Russian song. Directed by the director of Hardcore Henry, this narrative is more enjoyable when viewed in reverse (YT1) rather than in normal time (YT2).