user11962: This puzzle starts on move 19 of the below game. In order to get the puzzle correctly, it seems that you have to believe that black would not take the rook with 20.. gxf6. But I don't understand why this is – it's true that black will expose their king and breakup their pawn structure, but this seems like a small price to pay for taking the rook.
I ran it through the chess engine and it didn't seem to show any obvious traps that black would run into by taking the rook.

[FEN ""]
1. d4 d5 2. e4 dxe4 3. f3 { D00 Blackmar-Diemer Gambit: Gedult Gambit } Nc6 4. c3 Nf6 5. a3 e5 6. Be3 exd4 7. cxd4 exf3 8. Nxf3 Bg4 9. Bb5 a6 10. Bxc6+ bxc6 11. O-O Bd6 12. Qc2 Bxf3 13. Rxf3 Qd7 14. Nc3 O-O 15. Bg5 Be7 16. Raf1 Qxd4+ 17. Kh1 h6 18. Rf4 Qb6 19. Bxf6 Bxf6 20. Rxf6


user14079: To complete Ian Bush's answer, I find this line pretty:
1...gxf6 2.Ne4 Rfd8 3.Nxf6+ Kf8 4.Qh7 
[4...Ke7 5.Re1+ +- Ian Bush] 
4...Qf2!? [taking advantage of White's weak back rank]
5.Ng8+! 
[5.Qxh6+ Ke7 6.Qc1 Qd2 7.Qxc6 ought to be winning too, but it is not decisive yet] 
5...Ke6 
[5...Kf8 6.Nxh6+ Ke7 7.Qe5 Kf8 8.Re1 Qd4 9.Qe7+ Kg7 10.Nf5+; 
5...Rxg8 6.Qe5+ and 6.Rxf2] 
6.Qxh6+ f6 [The king cannot go to the f- nor the d-file because of Rxf2] 7.Qh3+ Kf7 8.Qb3+ and 9.Rxf2 since the Qb3 covers d1 !

user7726: After 20 ... gxf6 21 Ne4 black is in serious trouble. The threat is 22 Nxf6+ followed by 23 Qh7 mate, and in avoiding this she will have to make serious concessions. An example and not unreasonable line:
[Event ""]
[Site ""]
[Date "????.??.??"]
[Round ""]
[White ""]
[Black ""]
[Result "*"]
[FEN "r4rk1/2p2pp1/pqp2R1p/8/8/P1N5/1PQ3PP/5R1K b - - 0 1"]

1...gxf6 2.Ne4 Rfd8 3.Nxf6+ Kf8 4.Qh7 Ke7 5.Re1+ Kxf6 6.Qxh6+ Kf5 7.g4+ 
Kxg4 8.Qg7+ Kh4 9.Re4+ Kh5 10.Re5+ Kh4 11.Qg3# *

As a  just below 1900 player I can't claim to be able to see all the variations, but over the board Ne4 is obvious enough to give me enough worries not to take the rook.To see all the glorious details stick it in an engine. 

11962: <p><a href="https://lichess.org/training/63516" rel="nofollow noreferrer">This puzzle</a> starts on move 19 of the below game. In order to get the puzzle correctly, it seems that you have to believe that black would not take the rook with 20.. gxf6. But I don't understand why this is – it's true that black will expose their king and breakup their pawn structure, but this seems like a small price to pay for taking the rook.</p>

<p>I ran it through the chess engine and it didn't seem to show any obvious traps that black would run into by taking the rook.</p>

<hr>

<pre><code>[FEN ""]
1. d4 d5 2. e4 dxe4 3. f3 { D00 Blackmar-Diemer Gambit: Gedult Gambit } Nc6 4. c3 Nf6 5. a3 e5 6. Be3 exd4 7. cxd4 exf3 8. Nxf3 Bg4 9. Bb5 a6 10. Bxc6+ bxc6 11. O-O Bd6 12. Qc2 Bxf3 13. Rxf3 Qd7 14. Nc3 O-O 15. Bg5 Be7 16. Raf1 Qxd4+ 17. Kh1 h6 18. Rf4 Qb6 19. Bxf6 Bxf6 20. Rxf6
</code></pre>
 14079: <p>To complete Ian Bush's answer, I find this line pretty:</p>

<p>1...gxf6 2.Ne4 Rfd8 3.Nxf6+ Kf8 4.Qh7 </p>

<p>[4...Ke7 5.Re1+ +- Ian Bush] </p>

<p>4...Qf2!? [taking advantage of White's weak back rank]</p>

<p>5.Ng8+! 
[5.Qxh6+ Ke7 6.Qc1 Qd2 7.Qxc6 ought to be winning too, but it is not decisive yet] </p>

<p>5...Ke6 
[5...Kf8 6.Nxh6+ Ke7 7.Qe5 Kf8 8.Re1 Qd4 9.Qe7+ Kg7 10.Nf5+; 
5...Rxg8 6.Qe5+ and 6.Rxf2] </p>

<p>6.Qxh6+ f6 [The king cannot go to the f- nor the d-file because of Rxf2] 7.Qh3+ Kf7 8.Qb3+ and 9.Rxf2 since the Qb3 covers d1 !</p>
 7726: <p>After 20 ... gxf6 21 Ne4 black is in serious trouble. The threat is 22 Nxf6+ followed by 23 Qh7 mate, and in avoiding this she will have to make serious concessions. An example and not unreasonable line:</p>

<pre><code>[Event ""]
[Site ""]
[Date "????.??.??"]
[Round ""]
[White ""]
[Black ""]
[Result "*"]
[FEN "r4rk1/2p2pp1/pqp2R1p/8/8/P1N5/1PQ3PP/5R1K b - - 0 1"]

1...gxf6 2.Ne4 Rfd8 3.Nxf6+ Kf8 4.Qh7 Ke7 5.Re1+ Kxf6 6.Qxh6+ Kf5 7.g4+ 
Kxg4 8.Qg7+ Kh4 9.Re4+ Kh5 10.Re5+ Kh4 11.Qg3# *
</code></pre>

<p>As a  just below 1900 player I can't claim to be able to see all the variations, but over the board Ne4 is obvious enough to give me enough worries not to take the rook.To see all the glorious details stick it in an engine. </p>
