Before engines were introduced to chess in late 20th centry, there were many established openings that were played by the top players at that time. I am wondering if any of those openings were refuted (and thus are seldom played today) by engine? Namely, have engines found any flaws in established popular openings?
No, engines have not refuted any established openings, unless you're including nonsense openings like 1.e4 d6 2. Qg4, which humans already know to be unsound.
In 2012, Rybka's author Vasik Rajlich authored an April's Fool prank in Chessbase claiming to have proved the King's Gambit is at best a draw. Owning up to the prank a few days later, he said:
It's reasonable to construct a search tree of around 10^18 positions using modern technology. The chess alpha-beta tree is thought to have at least 10^45 positions. The alpha-beta tree for the King's Gambit will be at most 10x to 100x smaller than that. So, we're still probably a good 25 or so orders of magnitude away from being able to solve something like the King's Gambit. If processing power doubles every 18 months for the next century, we'll have the resources to do this around the year 2120, plus or minus a few decades.
The last couple of lines are the most relevant ones. It's 2019 as I write this, so if processing power doubles every 18 months for the next century, we're still a hundred years away from solving the King's Gambit, plus or minus a few decades. The same goes for every other opening, even those 20+ moves long. Talking in terms of order of magnitude, even if the alpha-beta tree for those openings are a billion times (10^9) smaller than chess's, we'd still be a good 16 orders of magnitude or so away from being able to solve it.
If you're also interested in endgames though, then there have been endgames which humans couldn't solve that have been solved by engines (or engines found flaws in human analysis).
I'm adding another answer because although this opening is not refuted, I believe it's quite close and OP will be interested.
Edit with better example: this line of the Najdorf, also played in the TCEC Season 18 superfinal, might very well be "busted".
[FEN ""] 1. e4 c5 2. Nf3 d6 3. d4 cxd4 4. Nxd4 Nf6 5. Nc3 a6 6. Bg5 e6 7. f4 Be7 8. Qf3 Qc7 9. O-O-O Nbd7 10. g4 b5 11. Bxf6 Nxf6 12. g5 Nd7 13. f5 Nc5 14. f6 gxf6 15. gxf6 Bf8 16. Rg1
This is another position that's been seen in loads of human games, but in this superfinal the engines might have found the refutation. When playing White, Stockfish reached +4 eval after four moves (16...h5 17. a3 Qb7 18. b4), while it reached +2 eval after only three moves as Black (16...h5 17. a3 Qb6). +2 is very near decisive; usually when Stockfish reaches this level it is expected to win against itself as White.
Caveat: Leela was not nearly as optimistic in the position. Still, it seems unlikely Leela is correct - after all it was not able to hold the game with Black.
Addendum: GM Matthew Sadler writes that as far as he knew prior to the game, this line was still playable, but based on the results it looks like future theory will consider these two games the refutation of 13...Nc5 (in other words, 13...Nc5 might be the losing move).